\(\int \frac {(a+b x^2)^2 (c+d x^2)^3}{(e+f x^2)^4} \, dx\) [235]

Optimal result

Integrand size = 28, antiderivative size = 426 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=-\frac {b d^2 (4 b d e-3 b c f-2 a d f) x}{f^5}+\frac {b^2 d^3 x^3}{3 f^4}-\frac {(b e-a f)^2 (d e-c f)^3 x}{6 e f^5 \left (e+f x^2\right )^3}+\frac {(b e-a f) (d e-c f)^2 (b e (25 d e-7 c f)-a f (13 d e+5 c f)) x}{24 e^2 f^5 \left (e+f x^2\right )^2}+\frac {(d e-c f) \left (2 a b e f \left (29 d^2 e^2-4 c d e f-c^2 f^2\right )-b^2 e^2 \left (55 d^2 e^2-32 c d e f+c^2 f^2\right )-a^2 f^2 \left (11 d^2 e^2+8 c d e f+5 c^2 f^2\right )\right ) x}{16 e^3 f^5 \left (e+f x^2\right )}-\frac {\left (2 a b e f \left (35 d^3 e^3-15 c d^2 e^2 f-3 c^2 d e f^2-c^3 f^3\right )-b^2 e^2 \left (105 d^3 e^3-105 c d^2 e^2 f+15 c^2 d e f^2+c^3 f^3\right )-a^2 f^2 \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{16 e^{7/2} f^{11/2}} \] Output:

-b*d^2*(-2*a*d*f-3*b*c*f+4*b*d*e)*x/f^5+1/3*b^2*d^3*x^3/f^4-1/6*(-a*f+b*e) 
^2*(-c*f+d*e)^3*x/e/f^5/(f*x^2+e)^3+1/24*(-a*f+b*e)*(-c*f+d*e)^2*(b*e*(-7* 
c*f+25*d*e)-a*f*(5*c*f+13*d*e))*x/e^2/f^5/(f*x^2+e)^2+1/16*(-c*f+d*e)*(2*a 
*b*e*f*(-c^2*f^2-4*c*d*e*f+29*d^2*e^2)-b^2*e^2*(c^2*f^2-32*c*d*e*f+55*d^2* 
e^2)-a^2*f^2*(5*c^2*f^2+8*c*d*e*f+11*d^2*e^2))*x/e^3/f^5/(f*x^2+e)-1/16*(2 
*a*b*e*f*(-c^3*f^3-3*c^2*d*e*f^2-15*c*d^2*e^2*f+35*d^3*e^3)-b^2*e^2*(c^3*f 
^3+15*c^2*d*e*f^2-105*c*d^2*e^2*f+105*d^3*e^3)-a^2*f^2*(5*c^3*f^3+3*c^2*d* 
e*f^2+3*c*d^2*e^2*f+5*d^3*e^3))*arctan(f^(1/2)*x/e^(1/2))/e^(7/2)/f^(11/2)
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 423, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\frac {-48 b d^2 \sqrt {f} (4 b d e-3 b c f-2 a d f) x+16 b^2 d^3 f^{3/2} x^3+\frac {8 \sqrt {f} (b e-a f)^2 (-d e+c f)^3 x}{e \left (e+f x^2\right )^3}+\frac {2 \sqrt {f} (-b e+a f) (d e-c f)^2 (a f (13 d e+5 c f)+b e (-25 d e+7 c f)) x}{e^2 \left (e+f x^2\right )^2}+\frac {3 \sqrt {f} (-d e+c f) \left (b^2 e^2 \left (55 d^2 e^2-32 c d e f+c^2 f^2\right )+2 a b e f \left (-29 d^2 e^2+4 c d e f+c^2 f^2\right )+a^2 f^2 \left (11 d^2 e^2+8 c d e f+5 c^2 f^2\right )\right ) x}{e^3 \left (e+f x^2\right )}+\frac {3 \left (2 a b e f \left (-35 d^3 e^3+15 c d^2 e^2 f+3 c^2 d e f^2+c^3 f^3\right )+b^2 e^2 \left (105 d^3 e^3-105 c d^2 e^2 f+15 c^2 d e f^2+c^3 f^3\right )+a^2 f^2 \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{e^{7/2}}}{48 f^{11/2}} \] Input:

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

Output:

(-48*b*d^2*Sqrt[f]*(4*b*d*e - 3*b*c*f - 2*a*d*f)*x + 16*b^2*d^3*f^(3/2)*x^ 
3 + (8*Sqrt[f]*(b*e - a*f)^2*(-(d*e) + c*f)^3*x)/(e*(e + f*x^2)^3) + (2*Sq 
rt[f]*(-(b*e) + a*f)*(d*e - c*f)^2*(a*f*(13*d*e + 5*c*f) + b*e*(-25*d*e + 
7*c*f))*x)/(e^2*(e + f*x^2)^2) + (3*Sqrt[f]*(-(d*e) + c*f)*(b^2*e^2*(55*d^ 
2*e^2 - 32*c*d*e*f + c^2*f^2) + 2*a*b*e*f*(-29*d^2*e^2 + 4*c*d*e*f + c^2*f 
^2) + a^2*f^2*(11*d^2*e^2 + 8*c*d*e*f + 5*c^2*f^2))*x)/(e^3*(e + f*x^2)) + 
 (3*(2*a*b*e*f*(-35*d^3*e^3 + 15*c*d^2*e^2*f + 3*c^2*d*e*f^2 + c^3*f^3) + 
b^2*e^2*(105*d^3*e^3 - 105*c*d^2*e^2*f + 15*c^2*d*e*f^2 + c^3*f^3) + a^2*f 
^2*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3))*ArcTan[(Sqrt[f 
]*x)/Sqrt[e]])/e^(7/2))/(48*f^(11/2))
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 704, normalized size of antiderivative = 1.65, 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, 25, 401, 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.

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

\(\Big \downarrow \) 425

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 299

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(b*(-1/4*((b*e - a*f)*x*(c + d*x^2)^3)/(e*f*(e + f*x^2)^2) + (-1/2*((b*e*( 
7*d*e - c*f) - 3*a*f*(d*e + c*f))*x*(c + d*x^2)^2)/(e*f*(e + f*x^2)) - (-1 
/3*(d*(b*e*(35*d*e - 3*c*f) - 3*a*f*(5*d*e + 3*c*f))*x*(c + d*x^2))/f + (- 
((d*(3*a*f*(15*d^2*e^2 - 4*c*d*e*f - 3*c^2*f^2) - b*e*(105*d^2*e^2 - 100*c 
*d*e*f + 3*c^2*f^2))*x)/f) - (3*(d*e - c*f)*(b*e*(35*d^2*e^2 - 10*c*d*e*f 
- c^2*f^2) - 3*a*f*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[f]*x)/S 
qrt[e]])/(Sqrt[e]*f^(3/2)))/(3*f))/(2*e*f))/(4*e*f)))/f - ((b*e - a*f)*(-1 
/6*((b*e - a*f)*x*(c + d*x^2)^3)/(e*f*(e + f*x^2)^3) + (-1/4*((b*e*(7*d*e 
- c*f) - a*f*(d*e + 5*c*f))*x*(c + d*x^2)^2)/(e*f*(e + f*x^2)^2) + (-1/2*( 
(b*e*(35*d^2*e^2 - 8*c*d*e*f - 3*c^2*f^2) - a*f*(5*d^2*e^2 + 4*c*d*e*f + 1 
5*c^2*f^2))*x*(c + d*x^2))/(e*f*(e + f*x^2)) - (-((d*(b*e*(105*d^2*e^2 - 1 
0*c*d*e*f - 3*c^2*f^2) - a*f*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2))*x)/f) 
 + (3*(b*e*(35*d^3*e^3 - 15*c*d^2*e^2*f - 3*c^2*d*e*f^2 - c^3*f^3) - a*f*( 
5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3))*ArcTan[(Sqrt[f]*x) 
/Sqrt[e]])/(Sqrt[e]*f^(3/2)))/(2*e*f))/(4*e*f))/(6*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 [A] (verified)

Time = 0.58 (sec) , antiderivative size = 724, normalized size of antiderivative = 1.70

method	result	size
default	\(\frac {b \,d^{2} \left (\frac {1}{3} b d f \,x^{3}+2 a d f x +3 b c f x -4 b d e x \right )}{f^{5}}+\frac {\frac {\frac {f^{2} \left (5 a^{2} c^{3} f^{5}+3 a^{2} c^{2} d e \,f^{4}+3 a^{2} c \,d^{2} e^{2} f^{3}-11 a^{2} d^{3} e^{3} f^{2}+2 a b \,c^{3} e \,f^{4}+6 a b \,c^{2} d \,e^{2} f^{3}-66 a b c \,d^{2} e^{3} f^{2}+58 a b \,d^{3} e^{4} f +b^{2} c^{3} e^{2} f^{3}-33 b^{2} c^{2} d \,e^{3} f^{2}+87 b^{2} c \,d^{2} e^{4} f -55 b^{2} d^{3} e^{5}\right ) x^{5}}{16 e^{3}}+\frac {f \left (5 a^{2} c^{3} f^{5}+3 a^{2} c^{2} d e \,f^{4}-3 a^{2} c \,d^{2} e^{2} f^{3}-5 a^{2} d^{3} e^{3} f^{2}+2 a b \,c^{3} e \,f^{4}-6 a b \,c^{2} d \,e^{2} f^{3}-30 a b c \,d^{2} e^{3} f^{2}+34 a b \,d^{3} e^{4} f -b^{2} c^{3} e^{2} f^{3}-15 b^{2} c^{2} d \,e^{3} f^{2}+51 b^{2} c \,d^{2} e^{4} f -35 b^{2} d^{3} e^{5}\right ) x^{3}}{6 e^{2}}+\frac {\left (11 a^{2} c^{3} f^{5}-3 a^{2} c^{2} d e \,f^{4}-3 a^{2} c \,d^{2} e^{2} f^{3}-5 a^{2} d^{3} e^{3} f^{2}-2 a b \,c^{3} e \,f^{4}-6 a b \,c^{2} d \,e^{2} f^{3}-30 a b c \,d^{2} e^{3} f^{2}+38 a b \,d^{3} e^{4} f -b^{2} c^{3} e^{2} f^{3}-15 b^{2} c^{2} d \,e^{3} f^{2}+57 b^{2} c \,d^{2} e^{4} f -41 b^{2} d^{3} e^{5}\right ) x}{16 e}}{\left (f \,x^{2}+e \right )^{3}}+\frac {\left (5 a^{2} c^{3} f^{5}+3 a^{2} c^{2} d e \,f^{4}+3 a^{2} c \,d^{2} e^{2} f^{3}+5 a^{2} d^{3} e^{3} f^{2}+2 a b \,c^{3} e \,f^{4}+6 a b \,c^{2} d \,e^{2} f^{3}+30 a b c \,d^{2} e^{3} f^{2}-70 a b \,d^{3} e^{4} f +b^{2} c^{3} e^{2} f^{3}+15 b^{2} c^{2} d \,e^{3} f^{2}-105 b^{2} c \,d^{2} e^{4} f +105 b^{2} d^{3} e^{5}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 e^{3} \sqrt {e f}}}{f^{5}}\)	\(724\)
risch	\(\text {Expression too large to display}\)	\(1291\)

Input:

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

Output:

b*d^2/f^5*(1/3*b*d*f*x^3+2*a*d*f*x+3*b*c*f*x-4*b*d*e*x)+1/f^5*((1/16*f^2*( 
5*a^2*c^3*f^5+3*a^2*c^2*d*e*f^4+3*a^2*c*d^2*e^2*f^3-11*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^3-66*a*b*c*d^2*e^3*f^2+58*a*b*d^3*e^4*f+b^2 
*c^3*e^2*f^3-33*b^2*c^2*d*e^3*f^2+87*b^2*c*d^2*e^4*f-55*b^2*d^3*e^5)/e^3*x 
^5+1/6*f*(5*a^2*c^3*f^5+3*a^2*c^2*d*e*f^4-3*a^2*c*d^2*e^2*f^3-5*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^3-30*a*b*c*d^2*e^3*f^2+34*a*b*d^3* 
e^4*f-b^2*c^3*e^2*f^3-15*b^2*c^2*d*e^3*f^2+51*b^2*c*d^2*e^4*f-35*b^2*d^3*e 
^5)/e^2*x^3+1/16*(11*a^2*c^3*f^5-3*a^2*c^2*d*e*f^4-3*a^2*c*d^2*e^2*f^3-5*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^3-30*a*b*c*d^2*e^3*f^2+38 
*a*b*d^3*e^4*f-b^2*c^3*e^2*f^3-15*b^2*c^2*d*e^3*f^2+57*b^2*c*d^2*e^4*f-41* 
b^2*d^3*e^5)/e*x)/(f*x^2+e)^3+1/16*(5*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+5*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+30*a*b 
*c*d^2*e^3*f^2-70*a*b*d^3*e^4*f+b^2*c^3*e^2*f^3+15*b^2*c^2*d*e^3*f^2-105*b 
^2*c*d^2*e^4*f+105*b^2*d^3*e^5)/e^3/(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. 1138 vs. \(2 (404) = 808\).

Time = 0.18 (sec) , antiderivative size = 2296, normalized size of antiderivative = 5.39 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/96*(32*b^2*d^3*e^4*f^5*x^9 - 96*(3*b^2*d^3*e^5*f^4 - (3*b^2*c*d^2 + 2*a 
*b*d^3)*e^4*f^5)*x^7 - 6*(231*b^2*d^3*e^6*f^3 - 5*a^2*c^3*e*f^8 - 77*(3*b^ 
2*c*d^2 + 2*a*b*d^3)*e^5*f^4 + 11*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^ 
4*f^5 - (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^3*f^6 - (2*a*b*c^3 + 3*a^2 
*c^2*d)*e^2*f^7)*x^5 - 16*(105*b^2*d^3*e^7*f^2 - 5*a^2*c^3*e^2*f^7 - 35*(3 
*b^2*c*d^2 + 2*a*b*d^3)*e^6*f^3 + 5*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)* 
e^5*f^4 + (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^4*f^5 - (2*a*b*c^3 + 3*a 
^2*c^2*d)*e^3*f^6)*x^3 - 3*(105*b^2*d^3*e^8 + 5*a^2*c^3*e^3*f^5 - 35*(3*b^ 
2*c*d^2 + 2*a*b*d^3)*e^7*f + 5*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^6*f 
^2 + (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^5*f^3 + (2*a*b*c^3 + 3*a^2*c^ 
2*d)*e^4*f^4 + (105*b^2*d^3*e^5*f^3 + 5*a^2*c^3*f^8 - 35*(3*b^2*c*d^2 + 2* 
a*b*d^3)*e^4*f^4 + 5*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^3*f^5 + (b^2* 
c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^2*f^6 + (2*a*b*c^3 + 3*a^2*c^2*d)*e*f^7 
)*x^6 + 3*(105*b^2*d^3*e^6*f^2 + 5*a^2*c^3*e*f^7 - 35*(3*b^2*c*d^2 + 2*a*b 
*d^3)*e^5*f^3 + 5*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^4*f^4 + (b^2*c^3 
 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^3*f^5 + (2*a*b*c^3 + 3*a^2*c^2*d)*e^2*f^6) 
*x^4 + 3*(105*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 + 5*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^5*f^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^2)*sqrt(-e*f)*log((f*x^2 - 2*sqrt(-e*f)*x - e)/(f*x^2 + e)) - 6*(105*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 829 vs. \(2 (404) = 808\).

Time = 0.13 (sec) , antiderivative size = 829, normalized size of antiderivative = 1.95 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\frac {{\left (105 \, b^{2} d^{3} e^{5} - 105 \, b^{2} c d^{2} e^{4} f - 70 \, 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} + b^{2} c^{3} e^{2} f^{3} + 6 \, a b c^{2} d e^{2} f^{3} + 3 \, 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} + 5 \, a^{2} c^{3} f^{5}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{16 \, \sqrt {e f} e^{3} f^{5}} - \frac {165 \, b^{2} d^{3} e^{5} f^{2} x^{5} - 261 \, b^{2} c d^{2} e^{4} f^{3} x^{5} - 174 \, a b d^{3} e^{4} f^{3} x^{5} + 99 \, b^{2} c^{2} d e^{3} f^{4} x^{5} + 198 \, a b c d^{2} e^{3} f^{4} x^{5} + 33 \, a^{2} d^{3} e^{3} f^{4} x^{5} - 3 \, b^{2} c^{3} e^{2} f^{5} x^{5} - 18 \, a b c^{2} d e^{2} f^{5} x^{5} - 9 \, a^{2} c d^{2} e^{2} f^{5} x^{5} - 6 \, a b c^{3} e f^{6} x^{5} - 9 \, a^{2} c^{2} d e f^{6} x^{5} - 15 \, a^{2} c^{3} f^{7} x^{5} + 280 \, b^{2} d^{3} e^{6} f x^{3} - 408 \, b^{2} c d^{2} e^{5} f^{2} x^{3} - 272 \, a b d^{3} e^{5} f^{2} x^{3} + 120 \, b^{2} c^{2} d e^{4} f^{3} x^{3} + 240 \, a b c d^{2} e^{4} f^{3} x^{3} + 40 \, a^{2} d^{3} e^{4} f^{3} x^{3} + 8 \, b^{2} c^{3} e^{3} f^{4} x^{3} + 48 \, a b c^{2} d e^{3} f^{4} x^{3} + 24 \, a^{2} c d^{2} e^{3} f^{4} x^{3} - 16 \, a b c^{3} e^{2} f^{5} x^{3} - 24 \, a^{2} c^{2} d e^{2} f^{5} x^{3} - 40 \, a^{2} c^{3} e f^{6} x^{3} + 123 \, b^{2} d^{3} e^{7} x - 171 \, b^{2} c d^{2} e^{6} f x - 114 \, a b d^{3} e^{6} f x + 45 \, b^{2} c^{2} d e^{5} f^{2} x + 90 \, a b c d^{2} e^{5} f^{2} x + 15 \, a^{2} d^{3} e^{5} f^{2} x + 3 \, b^{2} c^{3} e^{4} f^{3} x + 18 \, a b c^{2} d e^{4} f^{3} x + 9 \, a^{2} c d^{2} e^{4} f^{3} x + 6 \, a b c^{3} e^{3} f^{4} x + 9 \, a^{2} c^{2} d e^{3} f^{4} x - 33 \, a^{2} c^{3} e^{2} f^{5} x}{48 \, {\left (f x^{2} + e\right )}^{3} e^{3} f^{5}} + \frac {b^{2} d^{3} f^{8} x^{3} - 12 \, b^{2} d^{3} e f^{7} x + 9 \, b^{2} c d^{2} f^{8} x + 6 \, a b d^{3} f^{8} x}{3 \, f^{12}} \] Input:

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

Output:

1/16*(105*b^2*d^3*e^5 - 105*b^2*c*d^2*e^4*f - 70*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 + b^2*c^3*e^2*f^3 + 
 6*a*b*c^2*d*e^2*f^3 + 3*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 + 5*a^2*c^3*f^5)*arctan(f*x/sqrt(e*f))/(sqrt(e*f)*e^3*f^5) - 1/48*( 
165*b^2*d^3*e^5*f^2*x^5 - 261*b^2*c*d^2*e^4*f^3*x^5 - 174*a*b*d^3*e^4*f^3* 
x^5 + 99*b^2*c^2*d*e^3*f^4*x^5 + 198*a*b*c*d^2*e^3*f^4*x^5 + 33*a^2*d^3*e^ 
3*f^4*x^5 - 3*b^2*c^3*e^2*f^5*x^5 - 18*a*b*c^2*d*e^2*f^5*x^5 - 9*a^2*c*d^2 
*e^2*f^5*x^5 - 6*a*b*c^3*e*f^6*x^5 - 9*a^2*c^2*d*e*f^6*x^5 - 15*a^2*c^3*f^ 
7*x^5 + 280*b^2*d^3*e^6*f*x^3 - 408*b^2*c*d^2*e^5*f^2*x^3 - 272*a*b*d^3*e^ 
5*f^2*x^3 + 120*b^2*c^2*d*e^4*f^3*x^3 + 240*a*b*c*d^2*e^4*f^3*x^3 + 40*a^2 
*d^3*e^4*f^3*x^3 + 8*b^2*c^3*e^3*f^4*x^3 + 48*a*b*c^2*d*e^3*f^4*x^3 + 24*a 
^2*c*d^2*e^3*f^4*x^3 - 16*a*b*c^3*e^2*f^5*x^3 - 24*a^2*c^2*d*e^2*f^5*x^3 - 
 40*a^2*c^3*e*f^6*x^3 + 123*b^2*d^3*e^7*x - 171*b^2*c*d^2*e^6*f*x - 114*a* 
b*d^3*e^6*f*x + 45*b^2*c^2*d*e^5*f^2*x + 90*a*b*c*d^2*e^5*f^2*x + 15*a^2*d 
^3*e^5*f^2*x + 3*b^2*c^3*e^4*f^3*x + 18*a*b*c^2*d*e^4*f^3*x + 9*a^2*c*d^2* 
e^4*f^3*x + 6*a*b*c^3*e^3*f^4*x + 9*a^2*c^2*d*e^3*f^4*x - 33*a^2*c^3*e^2*f 
^5*x)/((f*x^2 + e)^3*e^3*f^5) + 1/3*(b^2*d^3*f^8*x^3 - 12*b^2*d^3*e*f^7*x 
+ 9*b^2*c*d^2*f^8*x + 6*a*b*d^3*f^8*x)/f^12
 

Mupad [B] (verification not implemented)

Time = 2.38 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {e}}\right )\,\left (5\,a^2\,c^3\,f^5+3\,a^2\,c^2\,d\,e\,f^4+3\,a^2\,c\,d^2\,e^2\,f^3+5\,a^2\,d^3\,e^3\,f^2+2\,a\,b\,c^3\,e\,f^4+6\,a\,b\,c^2\,d\,e^2\,f^3+30\,a\,b\,c\,d^2\,e^3\,f^2-70\,a\,b\,d^3\,e^4\,f+b^2\,c^3\,e^2\,f^3+15\,b^2\,c^2\,d\,e^3\,f^2-105\,b^2\,c\,d^2\,e^4\,f+105\,b^2\,d^3\,e^5\right )}{16\,e^{7/2}\,f^{11/2}}-\frac {\frac {x\,\left (-11\,a^2\,c^3\,f^5+3\,a^2\,c^2\,d\,e\,f^4+3\,a^2\,c\,d^2\,e^2\,f^3+5\,a^2\,d^3\,e^3\,f^2+2\,a\,b\,c^3\,e\,f^4+6\,a\,b\,c^2\,d\,e^2\,f^3+30\,a\,b\,c\,d^2\,e^3\,f^2-38\,a\,b\,d^3\,e^4\,f+b^2\,c^3\,e^2\,f^3+15\,b^2\,c^2\,d\,e^3\,f^2-57\,b^2\,c\,d^2\,e^4\,f+41\,b^2\,d^3\,e^5\right )}{16\,e}-\frac {x^5\,\left (5\,a^2\,c^3\,f^7+3\,a^2\,c^2\,d\,e\,f^6+3\,a^2\,c\,d^2\,e^2\,f^5-11\,a^2\,d^3\,e^3\,f^4+2\,a\,b\,c^3\,e\,f^6+6\,a\,b\,c^2\,d\,e^2\,f^5-66\,a\,b\,c\,d^2\,e^3\,f^4+58\,a\,b\,d^3\,e^4\,f^3+b^2\,c^3\,e^2\,f^5-33\,b^2\,c^2\,d\,e^3\,f^4+87\,b^2\,c\,d^2\,e^4\,f^3-55\,b^2\,d^3\,e^5\,f^2\right )}{16\,e^3}+\frac {x^3\,\left (-5\,a^2\,c^3\,f^6-3\,a^2\,c^2\,d\,e\,f^5+3\,a^2\,c\,d^2\,e^2\,f^4+5\,a^2\,d^3\,e^3\,f^3-2\,a\,b\,c^3\,e\,f^5+6\,a\,b\,c^2\,d\,e^2\,f^4+30\,a\,b\,c\,d^2\,e^3\,f^3-34\,a\,b\,d^3\,e^4\,f^2+b^2\,c^3\,e^2\,f^4+15\,b^2\,c^2\,d\,e^3\,f^3-51\,b^2\,c\,d^2\,e^4\,f^2+35\,b^2\,d^3\,e^5\,f\right )}{6\,e^2}}{e^3\,f^5+3\,e^2\,f^6\,x^2+3\,e\,f^7\,x^4+f^8\,x^6}-x\,\left (\frac {4\,b^2\,d^3\,e}{f^5}-\frac {b\,d^2\,\left (2\,a\,d+3\,b\,c\right )}{f^4}\right )+\frac {b^2\,d^3\,x^3}{3\,f^4} \] Input:

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

Output:

(atan((f^(1/2)*x)/e^(1/2))*(5*a^2*c^3*f^5 + 105*b^2*d^3*e^5 + 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 - 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 + 3*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 + 6*a*b*c^2*d*e^2*f^3))/(16*e^(7/2)*f^(11/2)) - ((x 
*(41*b^2*d^3*e^5 - 11*a^2*c^3*f^5 + 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 - 38*a*b*d^3*e^4*f + 3*a^2*c^2*d*e*f^4 - 57*b^2*c*d^2*e^4* 
f + 3*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 + 6* 
a*b*c^2*d*e^2*f^3))/(16*e) - (x^5*(5*a^2*c^3*f^7 - 11*a^2*d^3*e^3*f^4 + b^ 
2*c^3*e^2*f^5 - 55*b^2*d^3*e^5*f^2 + 2*a*b*c^3*e*f^6 + 58*a*b*d^3*e^4*f^3 
+ 3*a^2*c^2*d*e*f^6 + 3*a^2*c*d^2*e^2*f^5 + 87*b^2*c*d^2*e^4*f^3 - 33*b^2* 
c^2*d*e^3*f^4 - 66*a*b*c*d^2*e^3*f^4 + 6*a*b*c^2*d*e^2*f^5))/(16*e^3) + (x 
^3*(35*b^2*d^3*e^5*f - 5*a^2*c^3*f^6 + 5*a^2*d^3*e^3*f^3 + b^2*c^3*e^2*f^4 
 - 2*a*b*c^3*e*f^5 - 34*a*b*d^3*e^4*f^2 - 3*a^2*c^2*d*e*f^5 + 3*a^2*c*d^2* 
e^2*f^4 - 51*b^2*c*d^2*e^4*f^2 + 15*b^2*c^2*d*e^3*f^3 + 30*a*b*c*d^2*e^3*f 
^3 + 6*a*b*c^2*d*e^2*f^4))/(6*e^2))/(e^3*f^5 + f^8*x^6 + 3*e*f^7*x^4 + 3*e 
^2*f^6*x^2) - x*((4*b^2*d^3*e)/f^5 - (b*d^2*(2*a*d + 3*b*c))/f^4) + (b^2*d 
^3*x^3)/(3*f^4)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 2236, normalized size of antiderivative = 5.25 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(15*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**3*e**3*f**5 + 45 
*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**3*e**2*f**6*x**2 + 
45*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**3*e*f**7*x**4 + 1 
5*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**3*f**8*x**6 + 9*sq 
rt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*d*e**4*f**4 + 27*sqr 
t(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*d*e**3*f**5*x**2 + 27 
*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*d*e**2*f**6*x**4 
+ 9*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c**2*d*e*f**7*x**6 
+ 9*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d**2*e**5*f**3 + 
27*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d**2*e**4*f**4*x** 
2 + 27*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d**2*e**3*f**5 
*x**4 + 9*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*c*d**2*e**2*f 
**6*x**6 + 15*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**3*e**6 
*f**2 + 45*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**3*e**5*f* 
*3*x**2 + 45*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**3*e**4* 
f**4*x**4 + 15*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a**2*d**3*e** 
3*f**5*x**6 + 6*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**3*e** 
4*f**4 + 18*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**3*e**3*f* 
*5*x**2 + 18*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**3*e**2*f 
**6*x**4 + 6*sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*a*b*c**3*e*f...