\(\int \frac {(a+b x^2)^2 (c+d x^2)^3}{(e+f x^2)^3} \, dx\) [234]

Optimal result

Integrand size = 28, antiderivative size = 333 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\frac {d \left (a^2 d^2 f^2-6 a b d f (d e-c f)+3 b^2 \left (2 d^2 e^2-3 c d e f+c^2 f^2\right )\right ) x}{f^5}-\frac {b d^2 (3 b d e-3 b c f-2 a d f) x^3}{3 f^4}+\frac {b^2 d^3 x^5}{5 f^3}-\frac {(b e-a f)^2 (d e-c f)^3 x}{4 e f^5 \left (e+f x^2\right )^2}+\frac {(b e-a f) (d e-c f)^2 (b e (17 d e-5 c f)-3 a f (3 d e+c f)) x}{8 e^2 f^5 \left (e+f x^2\right )}+\frac {(d e-c f) \left (2 a b e f \left (35 d^2 e^2-10 c d e f-c^2 f^2\right )-3 b^2 e^2 \left (21 d^2 e^2-14 c d e f+c^2 f^2\right )-3 a^2 f^2 \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} f^{11/2}} \] Output:

d*(a^2*d^2*f^2-6*a*b*d*f*(-c*f+d*e)+3*b^2*(c^2*f^2-3*c*d*e*f+2*d^2*e^2))*x 
/f^5-1/3*b*d^2*(-2*a*d*f-3*b*c*f+3*b*d*e)*x^3/f^4+1/5*b^2*d^3*x^5/f^3-1/4* 
(-a*f+b*e)^2*(-c*f+d*e)^3*x/e/f^5/(f*x^2+e)^2+1/8*(-a*f+b*e)*(-c*f+d*e)^2* 
(b*e*(-5*c*f+17*d*e)-3*a*f*(c*f+3*d*e))*x/e^2/f^5/(f*x^2+e)+1/8*(-c*f+d*e) 
*(2*a*b*e*f*(-c^2*f^2-10*c*d*e*f+35*d^2*e^2)-3*b^2*e^2*(c^2*f^2-14*c*d*e*f 
+21*d^2*e^2)-3*a^2*f^2*(c^2*f^2+2*c*d*e*f+5*d^2*e^2))*arctan(f^(1/2)*x/e^( 
1/2))/e^(5/2)/f^(11/2)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\frac {d \left (a^2 d^2 f^2+6 a b d f (-d e+c f)+3 b^2 \left (2 d^2 e^2-3 c d e f+c^2 f^2\right )\right ) x}{f^5}-\frac {b d^2 (3 b d e-3 b c f-2 a d f) x^3}{3 f^4}+\frac {b^2 d^3 x^5}{5 f^3}-\frac {(b e-a f)^2 (d e-c f)^3 x}{4 e f^5 \left (e+f x^2\right )^2}+\frac {(b e-a f) (d e-c f)^2 (b e (17 d e-5 c f)-3 a f (3 d e+c f)) x}{8 e^2 f^5 \left (e+f x^2\right )}-\frac {(d e-c f) \left (3 b^2 e^2 \left (21 d^2 e^2-14 c d e f+c^2 f^2\right )+3 a^2 f^2 \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )+2 a b e f \left (-35 d^2 e^2+10 c d e f+c^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} f^{11/2}} \] Input:

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

Output:

(d*(a^2*d^2*f^2 + 6*a*b*d*f*(-(d*e) + c*f) + 3*b^2*(2*d^2*e^2 - 3*c*d*e*f 
+ c^2*f^2))*x)/f^5 - (b*d^2*(3*b*d*e - 3*b*c*f - 2*a*d*f)*x^3)/(3*f^4) + ( 
b^2*d^3*x^5)/(5*f^3) - ((b*e - a*f)^2*(d*e - c*f)^3*x)/(4*e*f^5*(e + f*x^2 
)^2) + ((b*e - a*f)*(d*e - c*f)^2*(b*e*(17*d*e - 5*c*f) - 3*a*f*(3*d*e + c 
*f))*x)/(8*e^2*f^5*(e + f*x^2)) - ((d*e - c*f)*(3*b^2*e^2*(21*d^2*e^2 - 14 
*c*d*e*f + c^2*f^2) + 3*a^2*f^2*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2) + 2*a*b* 
e*f*(-35*d^2*e^2 + 10*c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(8* 
e^(5/2)*f^(11/2))
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.76, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {425, 401, 25, 401, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(b*(-1/2*((b*e - a*f)*x*(c + d*x^2)^3)/(e*f*(e + f*x^2)) + ((d*(7*b*e - 5* 
a*f)*x*(c + d*x^2)^2)/(5*f) - ((d*(b*e*(35*d*e - 33*c*f) - 5*a*f*(5*d*e - 
3*c*f))*x*(c + d*x^2))/(3*f) + ((d*(5*a*f*(15*d^2*e^2 - 22*c*d*e*f + 3*c^2 
*f^2) - b*e*(105*d^2*e^2 - 190*c*d*e*f + 81*c^2*f^2))*x)/f + (15*(d*e - c* 
f)^2*(b*e*(7*d*e - c*f) - a*f*(5*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/ 
(Sqrt[e]*f^(3/2)))/(3*f))/(5*f))/(2*e*f)))/f - ((b*e - a*f)*(-1/4*((b*e - 
a*f)*x*(c + d*x^2)^3)/(e*f*(e + f*x^2)^2) + (-1/2*((b*e*(7*d*e - c*f) - 3* 
a*f*(d*e + c*f))*x*(c + d*x^2)^2)/(e*f*(e + f*x^2)) - (-1/3*(d*(b*e*(35*d* 
e - 3*c*f) - 3*a*f*(5*d*e + 3*c*f))*x*(c + d*x^2))/f + (-((d*(3*a*f*(15*d^ 
2*e^2 - 4*c*d*e*f - 3*c^2*f^2) - b*e*(105*d^2*e^2 - 100*c*d*e*f + 3*c^2*f^ 
2))*x)/f) - (3*(d*e - c*f)*(b*e*(35*d^2*e^2 - 10*c*d*e*f - c^2*f^2) - 3*a* 
f*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e] 
*f^(3/2)))/(3*f))/(2*e*f))/(4*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. \(645\) vs. \(2(315)=630\).

Time = 0.69 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.94

method	result	size
default	\(\frac {d \left (\frac {1}{5} f^{2} x^{5} b^{2} d^{2}+\frac {2}{3} a b \,d^{2} f^{2} x^{3}+b^{2} c d \,f^{2} x^{3}-b^{2} d^{2} e f \,x^{3}+a^{2} d^{2} f^{2} x +6 a b c d \,f^{2} x -6 a b \,d^{2} e f x +3 b^{2} c^{2} f^{2} x -9 b^{2} c d e f x +6 b^{2} d^{2} e^{2} x \right )}{f^{5}}+\frac {\frac {\frac {f \left (3 a^{2} c^{3} f^{5}+3 a^{2} c^{2} d e \,f^{4}-15 a^{2} c \,d^{2} e^{2} f^{3}+9 a^{2} d^{3} e^{3} f^{2}+2 a b \,c^{3} e \,f^{4}-30 a b \,c^{2} d \,e^{2} f^{3}+54 a b c \,d^{2} e^{3} f^{2}-26 a b \,d^{3} e^{4} f -5 b^{2} c^{3} e^{2} f^{3}+27 b^{2} c^{2} d \,e^{3} f^{2}-39 b^{2} c \,d^{2} e^{4} f +17 b^{2} d^{3} e^{5}\right ) x^{3}}{8 e^{2}}+\frac {\left (5 a^{2} c^{3} f^{5}-3 a^{2} c^{2} d e \,f^{4}-9 a^{2} c \,d^{2} e^{2} f^{3}+7 a^{2} d^{3} e^{3} f^{2}-2 a b \,c^{3} e \,f^{4}-18 a b \,c^{2} d \,e^{2} f^{3}+42 a b c \,d^{2} e^{3} f^{2}-22 a b \,d^{3} e^{4} f -3 b^{2} c^{3} e^{2} f^{3}+21 b^{2} c^{2} d \,e^{3} f^{2}-33 b^{2} c \,d^{2} e^{4} f +15 b^{2} d^{3} e^{5}\right ) x}{8 e}}{\left (f \,x^{2}+e \right )^{2}}+\frac {\left (3 a^{2} c^{3} f^{5}+3 a^{2} c^{2} d e \,f^{4}+9 a^{2} c \,d^{2} e^{2} f^{3}-15 a^{2} d^{3} e^{3} f^{2}+2 a b \,c^{3} e \,f^{4}+18 a b \,c^{2} d \,e^{2} f^{3}-90 a b c \,d^{2} e^{3} f^{2}+70 a b \,d^{3} e^{4} f +3 b^{2} c^{3} e^{2} f^{3}-45 b^{2} c^{2} d \,e^{3} f^{2}+105 b^{2} c \,d^{2} e^{4} f -63 b^{2} d^{3} e^{5}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 e^{2} \sqrt {e f}}}{f^{5}}\)	\(646\)
risch	\(\text {Expression too large to display}\)	\(1202\)

Input:

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

Output:

d/f^5*(1/5*f^2*x^5*b^2*d^2+2/3*a*b*d^2*f^2*x^3+b^2*c*d*f^2*x^3-b^2*d^2*e*f 
*x^3+a^2*d^2*f^2*x+6*a*b*c*d*f^2*x-6*a*b*d^2*e*f*x+3*b^2*c^2*f^2*x-9*b^2*c 
*d*e*f*x+6*b^2*d^2*e^2*x)+1/f^5*((1/8*f*(3*a^2*c^3*f^5+3*a^2*c^2*d*e*f^4-1 
5*a^2*c*d^2*e^2*f^3+9*a^2*d^3*e^3*f^2+2*a*b*c^3*e*f^4-30*a*b*c^2*d*e^2*f^3 
+54*a*b*c*d^2*e^3*f^2-26*a*b*d^3*e^4*f-5*b^2*c^3*e^2*f^3+27*b^2*c^2*d*e^3* 
f^2-39*b^2*c*d^2*e^4*f+17*b^2*d^3*e^5)/e^2*x^3+1/8*(5*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+7*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+42*a*b*c*d^2*e^3*f^2-22*a*b*d^3*e^4*f-3*b^2*c^3*e^2*f^3+21*b^2* 
c^2*d*e^3*f^2-33*b^2*c*d^2*e^4*f+15*b^2*d^3*e^5)/e*x)/(f*x^2+e)^2+1/8*(3*a 
^2*c^3*f^5+3*a^2*c^2*d*e*f^4+9*a^2*c*d^2*e^2*f^3-15*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^3-90*a*b*c*d^2*e^3*f^2+70*a*b*d^3*e^4*f+3*b^2 
*c^3*e^2*f^3-45*b^2*c^2*d*e^3*f^2+105*b^2*c*d^2*e^4*f-63*b^2*d^3*e^5)/e^2/ 
(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. 916 vs. \(2 (315) = 630\).

Time = 0.13 (sec) , antiderivative size = 1852, normalized size of antiderivative = 5.56 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/240*(48*b^2*d^3*e^3*f^5*x^9 - 16*(9*b^2*d^3*e^4*f^4 - 5*(3*b^2*c*d^2 + 
2*a*b*d^3)*e^3*f^5)*x^7 + 16*(63*b^2*d^3*e^5*f^3 - 35*(3*b^2*c*d^2 + 2*a*b 
*d^3)*e^4*f^4 + 15*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^3*f^5)*x^5 + 10 
*(315*b^2*d^3*e^6*f^2 + 9*a^2*c^3*e*f^7 - 175*(3*b^2*c*d^2 + 2*a*b*d^3)*e^ 
5*f^3 + 75*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^4*f^4 - 15*(b^2*c^3 + 6 
*a*b*c^2*d + 3*a^2*c*d^2)*e^3*f^5 + 3*(2*a*b*c^3 + 3*a^2*c^2*d)*e^2*f^6)*x 
^3 + 15*(63*b^2*d^3*e^7 - 3*a^2*c^3*e^2*f^5 - 35*(3*b^2*c*d^2 + 2*a*b*d^3) 
*e^6*f + 15*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^5*f^2 - 3*(b^2*c^3 + 6 
*a*b*c^2*d + 3*a^2*c*d^2)*e^4*f^3 - (2*a*b*c^3 + 3*a^2*c^2*d)*e^3*f^4 + (6 
3*b^2*d^3*e^5*f^2 - 3*a^2*c^3*f^7 - 35*(3*b^2*c*d^2 + 2*a*b*d^3)*e^4*f^3 + 
 15*(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 - (2*a*b*c^3 + 3*a^2*c^2*d)*e*f^6)*x^4 + 2*(63*b 
^2*d^3*e^6*f - 3*a^2*c^3*e*f^6 - 35*(3*b^2*c*d^2 + 2*a*b*d^3)*e^5*f^2 + 15 
*(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^2)*sqrt(-e*f 
)*log((f*x^2 - 2*sqrt(-e*f)*x - e)/(f*x^2 + e)) + 30*(63*b^2*d^3*e^7*f + 5 
*a^2*c^3*e^2*f^6 - 35*(3*b^2*c*d^2 + 2*a*b*d^3)*e^6*f^2 + 15*(3*b^2*c^2*d 
+ 6*a*b*c*d^2 + a^2*d^3)*e^5*f^3 - 3*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2) 
*e^4*f^4 - (2*a*b*c^3 + 3*a^2*c^2*d)*e^3*f^5)*x)/(e^3*f^8*x^4 + 2*e^4*f^7* 
x^2 + e^5*f^6), 1/120*(24*b^2*d^3*e^3*f^5*x^9 - 8*(9*b^2*d^3*e^4*f^4 - ...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1508 vs. \(2 (335) = 670\).

Time = 82.32 (sec) , antiderivative size = 1508, normalized size of antiderivative = 4.53 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

b**2*d**3*x**5/(5*f**3) + x**3*(2*a*b*d**3/(3*f**3) + b**2*c*d**2/f**3 - b 
**2*d**3*e/f**4) + x*(a**2*d**3/f**3 + 6*a*b*c*d**2/f**3 - 6*a*b*d**3*e/f* 
*4 + 3*b**2*c**2*d/f**3 - 9*b**2*c*d**2*e/f**4 + 6*b**2*d**3*e**2/f**5) - 
sqrt(-1/(e**5*f**11))*(c*f - d*e)*(3*a**2*c**2*f**4 + 6*a**2*c*d*e*f**3 + 
15*a**2*d**2*e**2*f**2 + 2*a*b*c**2*e*f**3 + 20*a*b*c*d*e**2*f**2 - 70*a*b 
*d**2*e**3*f + 3*b**2*c**2*e**2*f**2 - 42*b**2*c*d*e**3*f + 63*b**2*d**2*e 
**4)*log(-e**3*f**5*sqrt(-1/(e**5*f**11))*(c*f - d*e)*(3*a**2*c**2*f**4 + 
6*a**2*c*d*e*f**3 + 15*a**2*d**2*e**2*f**2 + 2*a*b*c**2*e*f**3 + 20*a*b*c* 
d*e**2*f**2 - 70*a*b*d**2*e**3*f + 3*b**2*c**2*e**2*f**2 - 42*b**2*c*d*e** 
3*f + 63*b**2*d**2*e**4)/(3*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 - 15*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 - 90*a*b*c*d**2*e**3*f**2 + 70*a*b*d**3*e**4*f + 3*b**2*c** 
3*e**2*f**3 - 45*b**2*c**2*d*e**3*f**2 + 105*b**2*c*d**2*e**4*f - 63*b**2* 
d**3*e**5) + x)/16 + sqrt(-1/(e**5*f**11))*(c*f - d*e)*(3*a**2*c**2*f**4 + 
 6*a**2*c*d*e*f**3 + 15*a**2*d**2*e**2*f**2 + 2*a*b*c**2*e*f**3 + 20*a*b*c 
*d*e**2*f**2 - 70*a*b*d**2*e**3*f + 3*b**2*c**2*e**2*f**2 - 42*b**2*c*d*e* 
*3*f + 63*b**2*d**2*e**4)*log(e**3*f**5*sqrt(-1/(e**5*f**11))*(c*f - d*e)* 
(3*a**2*c**2*f**4 + 6*a**2*c*d*e*f**3 + 15*a**2*d**2*e**2*f**2 + 2*a*b*c** 
2*e*f**3 + 20*a*b*c*d*e**2*f**2 - 70*a*b*d**2*e**3*f + 3*b**2*c**2*e**2*f* 
*2 - 42*b**2*c*d*e**3*f + 63*b**2*d**2*e**4)/(3*a**2*c**3*f**5 + 3*a**2...
 

Maxima [F(-2)]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (315) = 630\).

Time = 0.13 (sec) , antiderivative size = 709, normalized size of antiderivative = 2.13 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=-\frac {{\left (63 \, b^{2} d^{3} e^{5} - 105 \, b^{2} c d^{2} e^{4} f - 70 \, a b d^{3} e^{4} f + 45 \, b^{2} c^{2} d e^{3} f^{2} + 90 \, a b c d^{2} e^{3} f^{2} + 15 \, a^{2} d^{3} e^{3} f^{2} - 3 \, b^{2} c^{3} e^{2} f^{3} - 18 \, a b c^{2} d e^{2} f^{3} - 9 \, a^{2} c d^{2} e^{2} f^{3} - 2 \, a b c^{3} e f^{4} - 3 \, a^{2} c^{2} d e f^{4} - 3 \, a^{2} c^{3} f^{5}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \, \sqrt {e f} e^{2} f^{5}} + \frac {17 \, b^{2} d^{3} e^{5} f x^{3} - 39 \, b^{2} c d^{2} e^{4} f^{2} x^{3} - 26 \, a b d^{3} e^{4} f^{2} x^{3} + 27 \, b^{2} c^{2} d e^{3} f^{3} x^{3} + 54 \, a b c d^{2} e^{3} f^{3} x^{3} + 9 \, a^{2} d^{3} e^{3} f^{3} x^{3} - 5 \, b^{2} c^{3} e^{2} f^{4} x^{3} - 30 \, a b c^{2} d e^{2} f^{4} x^{3} - 15 \, a^{2} c d^{2} e^{2} f^{4} x^{3} + 2 \, a b c^{3} e f^{5} x^{3} + 3 \, a^{2} c^{2} d e f^{5} x^{3} + 3 \, a^{2} c^{3} f^{6} x^{3} + 15 \, b^{2} d^{3} e^{6} x - 33 \, b^{2} c d^{2} e^{5} f x - 22 \, a b d^{3} e^{5} f x + 21 \, b^{2} c^{2} d e^{4} f^{2} x + 42 \, a b c d^{2} e^{4} f^{2} x + 7 \, a^{2} d^{3} e^{4} f^{2} x - 3 \, b^{2} c^{3} e^{3} f^{3} x - 18 \, a b c^{2} d e^{3} f^{3} x - 9 \, a^{2} c d^{2} e^{3} f^{3} x - 2 \, a b c^{3} e^{2} f^{4} x - 3 \, a^{2} c^{2} d e^{2} f^{4} x + 5 \, a^{2} c^{3} e f^{5} x}{8 \, {\left (f x^{2} + e\right )}^{2} e^{2} f^{5}} + \frac {3 \, b^{2} d^{3} f^{12} x^{5} - 15 \, b^{2} d^{3} e f^{11} x^{3} + 15 \, b^{2} c d^{2} f^{12} x^{3} + 10 \, a b d^{3} f^{12} x^{3} + 90 \, b^{2} d^{3} e^{2} f^{10} x - 135 \, b^{2} c d^{2} e f^{11} x - 90 \, a b d^{3} e f^{11} x + 45 \, b^{2} c^{2} d f^{12} x + 90 \, a b c d^{2} f^{12} x + 15 \, a^{2} d^{3} f^{12} x}{15 \, f^{15}} \] Input:

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

Output:

-1/8*(63*b^2*d^3*e^5 - 105*b^2*c*d^2*e^4*f - 70*a*b*d^3*e^4*f + 45*b^2*c^2 
*d*e^3*f^2 + 90*a*b*c*d^2*e^3*f^2 + 15*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 - 3*a^2*c^3*f^5)*arctan(f*x/sqrt(e*f))/(sqrt(e*f)*e^2*f^5) + 1/8 
*(17*b^2*d^3*e^5*f*x^3 - 39*b^2*c*d^2*e^4*f^2*x^3 - 26*a*b*d^3*e^4*f^2*x^3 
 + 27*b^2*c^2*d*e^3*f^3*x^3 + 54*a*b*c*d^2*e^3*f^3*x^3 + 9*a^2*d^3*e^3*f^3 
*x^3 - 5*b^2*c^3*e^2*f^4*x^3 - 30*a*b*c^2*d*e^2*f^4*x^3 - 15*a^2*c*d^2*e^2 
*f^4*x^3 + 2*a*b*c^3*e*f^5*x^3 + 3*a^2*c^2*d*e*f^5*x^3 + 3*a^2*c^3*f^6*x^3 
 + 15*b^2*d^3*e^6*x - 33*b^2*c*d^2*e^5*f*x - 22*a*b*d^3*e^5*f*x + 21*b^2*c 
^2*d*e^4*f^2*x + 42*a*b*c*d^2*e^4*f^2*x + 7*a^2*d^3*e^4*f^2*x - 3*b^2*c^3* 
e^3*f^3*x - 18*a*b*c^2*d*e^3*f^3*x - 9*a^2*c*d^2*e^3*f^3*x - 2*a*b*c^3*e^2 
*f^4*x - 3*a^2*c^2*d*e^2*f^4*x + 5*a^2*c^3*e*f^5*x)/((f*x^2 + e)^2*e^2*f^5 
) + 1/15*(3*b^2*d^3*f^12*x^5 - 15*b^2*d^3*e*f^11*x^3 + 15*b^2*c*d^2*f^12*x 
^3 + 10*a*b*d^3*f^12*x^3 + 90*b^2*d^3*e^2*f^10*x - 135*b^2*c*d^2*e*f^11*x 
- 90*a*b*d^3*e*f^11*x + 45*b^2*c^2*d*f^12*x + 90*a*b*c*d^2*f^12*x + 15*a^2 
*d^3*f^12*x)/f^15
 

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 901, normalized size of antiderivative = 2.71 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx=x\,\left (\frac {a^2\,d^3+6\,a\,b\,c\,d^2+3\,b^2\,c^2\,d}{f^3}+\frac {3\,e\,\left (\frac {3\,b^2\,d^3\,e}{f^4}-\frac {b\,d^2\,\left (2\,a\,d+3\,b\,c\right )}{f^3}\right )}{f}-\frac {3\,b^2\,d^3\,e^2}{f^5}\right )-\frac {\frac {x\,\left (-5\,a^2\,c^3\,f^5+3\,a^2\,c^2\,d\,e\,f^4+9\,a^2\,c\,d^2\,e^2\,f^3-7\,a^2\,d^3\,e^3\,f^2+2\,a\,b\,c^3\,e\,f^4+18\,a\,b\,c^2\,d\,e^2\,f^3-42\,a\,b\,c\,d^2\,e^3\,f^2+22\,a\,b\,d^3\,e^4\,f+3\,b^2\,c^3\,e^2\,f^3-21\,b^2\,c^2\,d\,e^3\,f^2+33\,b^2\,c\,d^2\,e^4\,f-15\,b^2\,d^3\,e^5\right )}{8\,e}-\frac {x^3\,\left (3\,a^2\,c^3\,f^6+3\,a^2\,c^2\,d\,e\,f^5-15\,a^2\,c\,d^2\,e^2\,f^4+9\,a^2\,d^3\,e^3\,f^3+2\,a\,b\,c^3\,e\,f^5-30\,a\,b\,c^2\,d\,e^2\,f^4+54\,a\,b\,c\,d^2\,e^3\,f^3-26\,a\,b\,d^3\,e^4\,f^2-5\,b^2\,c^3\,e^2\,f^4+27\,b^2\,c^2\,d\,e^3\,f^3-39\,b^2\,c\,d^2\,e^4\,f^2+17\,b^2\,d^3\,e^5\,f\right )}{8\,e^2}}{e^2\,f^5+2\,e\,f^6\,x^2+f^7\,x^4}-x^3\,\left (\frac {b^2\,d^3\,e}{f^4}-\frac {b\,d^2\,\left (2\,a\,d+3\,b\,c\right )}{3\,f^3}\right )+\frac {b^2\,d^3\,x^5}{5\,f^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x\,\left (c\,f-d\,e\right )\,\left (3\,a^2\,c^2\,f^4+6\,a^2\,c\,d\,e\,f^3+15\,a^2\,d^2\,e^2\,f^2+2\,a\,b\,c^2\,e\,f^3+20\,a\,b\,c\,d\,e^2\,f^2-70\,a\,b\,d^2\,e^3\,f+3\,b^2\,c^2\,e^2\,f^2-42\,b^2\,c\,d\,e^3\,f+63\,b^2\,d^2\,e^4\right )}{\sqrt {e}\,\left (3\,a^2\,c^3\,f^5+3\,a^2\,c^2\,d\,e\,f^4+9\,a^2\,c\,d^2\,e^2\,f^3-15\,a^2\,d^3\,e^3\,f^2+2\,a\,b\,c^3\,e\,f^4+18\,a\,b\,c^2\,d\,e^2\,f^3-90\,a\,b\,c\,d^2\,e^3\,f^2+70\,a\,b\,d^3\,e^4\,f+3\,b^2\,c^3\,e^2\,f^3-45\,b^2\,c^2\,d\,e^3\,f^2+105\,b^2\,c\,d^2\,e^4\,f-63\,b^2\,d^3\,e^5\right )}\right )\,\left (c\,f-d\,e\right )\,\left (3\,a^2\,c^2\,f^4+6\,a^2\,c\,d\,e\,f^3+15\,a^2\,d^2\,e^2\,f^2+2\,a\,b\,c^2\,e\,f^3+20\,a\,b\,c\,d\,e^2\,f^2-70\,a\,b\,d^2\,e^3\,f+3\,b^2\,c^2\,e^2\,f^2-42\,b^2\,c\,d\,e^3\,f+63\,b^2\,d^2\,e^4\right )}{8\,e^{5/2}\,f^{11/2}} \] Input:

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

Output:

x*((a^2*d^3 + 3*b^2*c^2*d + 6*a*b*c*d^2)/f^3 + (3*e*((3*b^2*d^3*e)/f^4 - ( 
b*d^2*(2*a*d + 3*b*c))/f^3))/f - (3*b^2*d^3*e^2)/f^5) - ((x*(3*b^2*c^3*e^2 
*f^3 - 15*b^2*d^3*e^5 - 7*a^2*d^3*e^3*f^2 - 5*a^2*c^3*f^5 + 2*a*b*c^3*e*f^ 
4 + 22*a*b*d^3*e^4*f + 3*a^2*c^2*d*e*f^4 + 33*b^2*c*d^2*e^4*f + 9*a^2*c*d^ 
2*e^2*f^3 - 21*b^2*c^2*d*e^3*f^2 - 42*a*b*c*d^2*e^3*f^2 + 18*a*b*c^2*d*e^2 
*f^3))/(8*e) - (x^3*(3*a^2*c^3*f^6 + 17*b^2*d^3*e^5*f + 9*a^2*d^3*e^3*f^3 
- 5*b^2*c^3*e^2*f^4 + 2*a*b*c^3*e*f^5 - 26*a*b*d^3*e^4*f^2 + 3*a^2*c^2*d*e 
*f^5 - 15*a^2*c*d^2*e^2*f^4 - 39*b^2*c*d^2*e^4*f^2 + 27*b^2*c^2*d*e^3*f^3 
+ 54*a*b*c*d^2*e^3*f^3 - 30*a*b*c^2*d*e^2*f^4))/(8*e^2))/(e^2*f^5 + f^7*x^ 
4 + 2*e*f^6*x^2) - x^3*((b^2*d^3*e)/f^4 - (b*d^2*(2*a*d + 3*b*c))/(3*f^3)) 
 + (b^2*d^3*x^5)/(5*f^3) + (atan((f^(1/2)*x*(c*f - d*e)*(3*a^2*c^2*f^4 + 6 
3*b^2*d^2*e^4 + 15*a^2*d^2*e^2*f^2 + 3*b^2*c^2*e^2*f^2 + 2*a*b*c^2*e*f^3 - 
 70*a*b*d^2*e^3*f + 6*a^2*c*d*e*f^3 - 42*b^2*c*d*e^3*f + 20*a*b*c*d*e^2*f^ 
2))/(e^(1/2)*(3*a^2*c^3*f^5 - 63*b^2*d^3*e^5 - 15*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 + 70*a*b*d^3*e^4*f + 3*a^2*c^2*d*e*f^4 + 105 
*b^2*c*d^2*e^4*f + 9*a^2*c*d^2*e^2*f^3 - 45*b^2*c^2*d*e^3*f^2 - 90*a*b*c*d 
^2*e^3*f^2 + 18*a*b*c^2*d*e^2*f^3)))*(c*f - d*e)*(3*a^2*c^2*f^4 + 63*b^2*d 
^2*e^4 + 15*a^2*d^2*e^2*f^2 + 3*b^2*c^2*e^2*f^2 + 2*a*b*c^2*e*f^3 - 70*a*b 
*d^2*e^3*f + 6*a^2*c*d*e*f^3 - 42*b^2*c*d*e^3*f + 20*a*b*c*d*e^2*f^2))/(8* 
e^(5/2)*f^(11/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 1724, normalized size of antiderivative = 5.18 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(45*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**3*e**2*f**5 + 90 
*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**3*e*f**6*x**2 + 45* 
sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**3*f**7*x**4 + 45*sqr 
t(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*d*e**3*f**4 + 90*sqrt 
(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*d*e**2*f**5*x**2 + 45* 
sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*d*e*f**6*x**4 + 13 
5*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d**2*e**4*f**3 + 27 
0*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d**2*e**3*f**4*x**2 
 + 135*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d**2*e**2*f**5 
*x**4 - 225*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**3*e**5*f 
**2 - 450*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**3*e**4*f** 
3*x**2 - 225*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**3*e**3* 
f**4*x**4 + 30*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**3*e**3 
*f**4 + 60*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**3*e**2*f** 
5*x**2 + 30*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**3*e*f**6* 
x**4 + 270*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*d*e**4*f 
**3 + 540*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*d*e**3*f* 
*4*x**2 + 270*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**2*d*e** 
2*f**5*x**4 - 1350*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c*d** 
2*e**5*f**2 - 2700*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c*...