\(\int (a+b x^2)^2 (c+d x^2)^2 (e+f x^2)^3 \, dx\) [225]

Optimal result

Integrand size = 28, antiderivative size = 362 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=a^2 c^2 e^3 x+\frac {1}{3} a c e^2 (2 b c e+2 a d e+3 a c f) x^3+\frac {1}{5} e \left (b^2 c^2 e^2+2 a b c e (2 d e+3 c f)+a^2 \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (b^2 c e^2 (2 d e+3 c f)+a^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+2 a b e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^7+\frac {1}{9} \left (a^2 d f^2 (3 d e+2 c f)+2 a b f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+b^2 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^9+\frac {1}{11} f \left (a^2 d^2 f^2+2 a b d f (3 d e+2 c f)+b^2 \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x^{11}+\frac {1}{13} b d f^2 (3 b d e+2 b c f+2 a d f) x^{13}+\frac {1}{15} b^2 d^2 f^3 x^{15} \] Output:

a^2*c^2*e^3*x+1/3*a*c*e^2*(3*a*c*f+2*a*d*e+2*b*c*e)*x^3+1/5*e*(b^2*c^2*e^2 
+2*a*b*c*e*(3*c*f+2*d*e)+a^2*(3*c^2*f^2+6*c*d*e*f+d^2*e^2))*x^5+1/7*(b^2*c 
*e^2*(3*c*f+2*d*e)+a^2*f*(c^2*f^2+6*c*d*e*f+3*d^2*e^2)+2*a*b*e*(3*c^2*f^2+ 
6*c*d*e*f+d^2*e^2))*x^7+1/9*(a^2*d*f^2*(2*c*f+3*d*e)+2*a*b*f*(c^2*f^2+6*c* 
d*e*f+3*d^2*e^2)+b^2*e*(3*c^2*f^2+6*c*d*e*f+d^2*e^2))*x^9+1/11*f*(a^2*d^2* 
f^2+2*a*b*d*f*(2*c*f+3*d*e)+b^2*(c^2*f^2+6*c*d*e*f+3*d^2*e^2))*x^11+1/13*b 
*d*f^2*(2*a*d*f+2*b*c*f+3*b*d*e)*x^13+1/15*b^2*d^2*f^3*x^15
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=a^2 c^2 e^3 x+\frac {1}{3} a c e^2 (2 b c e+2 a d e+3 a c f) x^3+\frac {1}{5} e \left (b^2 c^2 e^2+2 a b c e (2 d e+3 c f)+a^2 \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (b^2 c e^2 (2 d e+3 c f)+a^2 f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+2 a b e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^7+\frac {1}{9} \left (a^2 d f^2 (3 d e+2 c f)+2 a b f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+b^2 e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^9+\frac {1}{11} f \left (a^2 d^2 f^2+2 a b d f (3 d e+2 c f)+b^2 \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x^{11}+\frac {1}{13} b d f^2 (3 b d e+2 b c f+2 a d f) x^{13}+\frac {1}{15} b^2 d^2 f^3 x^{15} \] Input:

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

Output:

a^2*c^2*e^3*x + (a*c*e^2*(2*b*c*e + 2*a*d*e + 3*a*c*f)*x^3)/3 + (e*(b^2*c^ 
2*e^2 + 2*a*b*c*e*(2*d*e + 3*c*f) + a^2*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2)) 
*x^5)/5 + ((b^2*c*e^2*(2*d*e + 3*c*f) + a^2*f*(3*d^2*e^2 + 6*c*d*e*f + c^2 
*f^2) + 2*a*b*e*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^7)/7 + ((a^2*d*f^2*(3 
*d*e + 2*c*f) + 2*a*b*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) + b^2*e*(d^2*e^2 
 + 6*c*d*e*f + 3*c^2*f^2))*x^9)/9 + (f*(a^2*d^2*f^2 + 2*a*b*d*f*(3*d*e + 2 
*c*f) + b^2*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2))*x^11)/11 + (b*d*f^2*(3*b*d* 
e + 2*b*c*f + 2*a*d*f)*x^13)/13 + (b^2*d^2*f^3*x^15)/15
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {396, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 396

\(\displaystyle \int \left (f x^{10} \left (a^2 d^2 f^2+2 a b d f (2 c f+3 d e)+b^2 \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )\right )+x^8 \left (a^2 d f^2 (2 c f+3 d e)+2 a b f \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+b^2 e \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )\right )+x^6 \left (a^2 f \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+2 a b e \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+b^2 c e^2 (3 c f+2 d e)\right )+e x^4 \left (a^2 \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+2 a b c e (3 c f+2 d e)+b^2 c^2 e^2\right )+a^2 c^2 e^3+a c e^2 x^2 (3 a c f+2 a d e+2 b c e)+b d f^2 x^{12} (2 a d f+2 b c f+3 b d e)+b^2 d^2 f^3 x^{14}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{11} f x^{11} \left (a^2 d^2 f^2+2 a b d f (2 c f+3 d e)+b^2 \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )\right )+\frac {1}{9} x^9 \left (a^2 d f^2 (2 c f+3 d e)+2 a b f \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+b^2 e \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )\right )+\frac {1}{7} x^7 \left (a^2 f \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+2 a b e \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+b^2 c e^2 (3 c f+2 d e)\right )+\frac {1}{5} e x^5 \left (a^2 \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+2 a b c e (3 c f+2 d e)+b^2 c^2 e^2\right )+a^2 c^2 e^3 x+\frac {1}{3} a c e^2 x^3 (3 a c f+2 a d e+2 b c e)+\frac {1}{13} b d f^2 x^{13} (2 a d f+2 b c f+3 b d e)+\frac {1}{15} b^2 d^2 f^3 x^{15}\)

Input:

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

Output:

a^2*c^2*e^3*x + (a*c*e^2*(2*b*c*e + 2*a*d*e + 3*a*c*f)*x^3)/3 + (e*(b^2*c^ 
2*e^2 + 2*a*b*c*e*(2*d*e + 3*c*f) + a^2*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2)) 
*x^5)/5 + ((b^2*c*e^2*(2*d*e + 3*c*f) + a^2*f*(3*d^2*e^2 + 6*c*d*e*f + c^2 
*f^2) + 2*a*b*e*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^7)/7 + ((a^2*d*f^2*(3 
*d*e + 2*c*f) + 2*a*b*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) + b^2*e*(d^2*e^2 
 + 6*c*d*e*f + 3*c^2*f^2))*x^9)/9 + (f*(a^2*d^2*f^2 + 2*a*b*d*f*(3*d*e + 2 
*c*f) + b^2*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2))*x^11)/11 + (b*d*f^2*(3*b*d* 
e + 2*b*c*f + 2*a*d*f)*x^13)/13 + (b^2*d^2*f^3*x^15)/15
 

Defintions of rubi rules used

rule 396

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2)^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q* 
(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ 
[q, 0] && IGtQ[r, 0]
 

rule 2009

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

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.09

method	result	size
default	\(\frac {b^{2} d^{2} f^{3} x^{15}}{15}+\frac {\left (\left (2 a b \,d^{2}+2 b^{2} c d \right ) f^{3}+3 b^{2} d^{2} e \,f^{2}\right ) x^{13}}{13}+\frac {\left (\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) f^{3}+3 \left (2 a b \,d^{2}+2 b^{2} c d \right ) e \,f^{2}+3 b^{2} d^{2} e^{2} f \right ) x^{11}}{11}+\frac {\left (\left (2 a^{2} c d +2 b \,c^{2} a \right ) f^{3}+3 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) e \,f^{2}+3 \left (2 a b \,d^{2}+2 b^{2} c d \right ) e^{2} f +b^{2} d^{2} e^{3}\right ) x^{9}}{9}+\frac {\left (a^{2} c^{2} f^{3}+3 \left (2 a^{2} c d +2 b \,c^{2} a \right ) e \,f^{2}+3 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) e^{2} f +\left (2 a b \,d^{2}+2 b^{2} c d \right ) e^{3}\right ) x^{7}}{7}+\frac {\left (3 a^{2} c^{2} e \,f^{2}+3 \left (2 a^{2} c d +2 b \,c^{2} a \right ) e^{2} f +\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) e^{3}\right ) x^{5}}{5}+\frac {\left (3 a^{2} c^{2} e^{2} f +\left (2 a^{2} c d +2 b \,c^{2} a \right ) e^{3}\right ) x^{3}}{3}+a^{2} c^{2} e^{3} x\)	\(395\)
norman	\(a^{2} c^{2} e^{3} x +\left (a^{2} c^{2} e^{2} f +\frac {2}{3} a^{2} c d \,e^{3}+\frac {2}{3} a b \,c^{2} e^{3}\right ) x^{3}+\left (\frac {3}{5} a^{2} c^{2} e \,f^{2}+\frac {6}{5} a^{2} c d \,e^{2} f +\frac {1}{5} a^{2} d^{2} e^{3}+\frac {6}{5} a b \,c^{2} e^{2} f +\frac {4}{5} a b c d \,e^{3}+\frac {1}{5} b^{2} c^{2} e^{3}\right ) x^{5}+\left (\frac {1}{7} a^{2} c^{2} f^{3}+\frac {6}{7} a^{2} c d e \,f^{2}+\frac {3}{7} a^{2} d^{2} e^{2} f +\frac {6}{7} a b \,c^{2} e \,f^{2}+\frac {12}{7} a b c d \,e^{2} f +\frac {2}{7} a b \,d^{2} e^{3}+\frac {3}{7} b^{2} c^{2} e^{2} f +\frac {2}{7} b^{2} d \,e^{3} c \right ) x^{7}+\left (\frac {2}{9} a^{2} c d \,f^{3}+\frac {1}{3} a^{2} d^{2} e \,f^{2}+\frac {2}{9} a b \,c^{2} f^{3}+\frac {4}{3} a b c d e \,f^{2}+\frac {2}{3} a b \,d^{2} e^{2} f +\frac {1}{3} b^{2} c^{2} e \,f^{2}+\frac {2}{3} b^{2} c d \,e^{2} f +\frac {1}{9} b^{2} d^{2} e^{3}\right ) x^{9}+\left (\frac {1}{11} a^{2} d^{2} f^{3}+\frac {4}{11} a b c d \,f^{3}+\frac {6}{11} a b \,d^{2} e \,f^{2}+\frac {1}{11} b^{2} c^{2} f^{3}+\frac {6}{11} b^{2} c d e \,f^{2}+\frac {3}{11} b^{2} d^{2} e^{2} f \right ) x^{11}+\left (\frac {2}{13} a b \,d^{2} f^{3}+\frac {2}{13} b^{2} c d \,f^{3}+\frac {3}{13} b^{2} d^{2} e \,f^{2}\right ) x^{13}+\frac {b^{2} d^{2} f^{3} x^{15}}{15}\)	\(424\)
gosper	\(\frac {6}{5} x^{5} a^{2} c d \,e^{2} f +x^{3} a^{2} c^{2} e^{2} f +\frac {2}{3} x^{3} a^{2} c d \,e^{3}+\frac {1}{5} x^{5} a^{2} d^{2} e^{3}+\frac {3}{5} x^{5} a^{2} c^{2} e \,f^{2}+\frac {3}{7} x^{7} a^{2} d^{2} e^{2} f +\frac {1}{5} x^{5} b^{2} c^{2} e^{3}+\frac {1}{7} x^{7} a^{2} c^{2} f^{3}+\frac {6}{5} x^{5} a b \,c^{2} e^{2} f +\frac {3}{11} x^{11} b^{2} d^{2} e^{2} f +\frac {2}{13} x^{13} a b \,d^{2} f^{3}+\frac {2}{13} x^{13} b^{2} c d \,f^{3}+\frac {3}{13} x^{13} b^{2} d^{2} e \,f^{2}+\frac {2}{7} x^{7} a b \,d^{2} e^{3}+\frac {3}{7} x^{7} b^{2} c^{2} e^{2} f +\frac {2}{7} x^{7} b^{2} d \,e^{3} c +\frac {2}{9} x^{9} a^{2} c d \,f^{3}+\frac {1}{3} x^{9} a^{2} d^{2} e \,f^{2}+\frac {2}{9} x^{9} a b \,c^{2} f^{3}+\frac {4}{5} x^{5} a b c d \,e^{3}+\frac {6}{7} x^{7} a^{2} c d e \,f^{2}+\frac {2}{3} x^{3} a b \,c^{2} e^{3}+\frac {2}{3} x^{9} b^{2} c d \,e^{2} f +\frac {6}{7} x^{7} a b \,c^{2} e \,f^{2}+\frac {2}{3} x^{9} a b \,d^{2} e^{2} f +\frac {1}{3} x^{9} b^{2} c^{2} e \,f^{2}+\frac {4}{3} x^{9} a b c d e \,f^{2}+\frac {6}{11} x^{11} b^{2} c d e \,f^{2}+\frac {1}{11} x^{11} b^{2} c^{2} f^{3}+\frac {1}{9} x^{9} b^{2} d^{2} e^{3}+\frac {1}{11} x^{11} a^{2} d^{2} f^{3}+\frac {12}{7} x^{7} a b c d \,e^{2} f +\frac {6}{11} x^{11} a b \,d^{2} e \,f^{2}+\frac {4}{11} x^{11} a b c d \,f^{3}+a^{2} c^{2} e^{3} x +\frac {1}{15} b^{2} d^{2} f^{3} x^{15}\)	\(496\)
risch	\(\frac {6}{5} x^{5} a^{2} c d \,e^{2} f +x^{3} a^{2} c^{2} e^{2} f +\frac {2}{3} x^{3} a^{2} c d \,e^{3}+\frac {1}{5} x^{5} a^{2} d^{2} e^{3}+\frac {3}{5} x^{5} a^{2} c^{2} e \,f^{2}+\frac {3}{7} x^{7} a^{2} d^{2} e^{2} f +\frac {1}{5} x^{5} b^{2} c^{2} e^{3}+\frac {1}{7} x^{7} a^{2} c^{2} f^{3}+\frac {6}{5} x^{5} a b \,c^{2} e^{2} f +\frac {3}{11} x^{11} b^{2} d^{2} e^{2} f +\frac {2}{13} x^{13} a b \,d^{2} f^{3}+\frac {2}{13} x^{13} b^{2} c d \,f^{3}+\frac {3}{13} x^{13} b^{2} d^{2} e \,f^{2}+\frac {2}{7} x^{7} a b \,d^{2} e^{3}+\frac {3}{7} x^{7} b^{2} c^{2} e^{2} f +\frac {2}{7} x^{7} b^{2} d \,e^{3} c +\frac {2}{9} x^{9} a^{2} c d \,f^{3}+\frac {1}{3} x^{9} a^{2} d^{2} e \,f^{2}+\frac {2}{9} x^{9} a b \,c^{2} f^{3}+\frac {4}{5} x^{5} a b c d \,e^{3}+\frac {6}{7} x^{7} a^{2} c d e \,f^{2}+\frac {2}{3} x^{3} a b \,c^{2} e^{3}+\frac {2}{3} x^{9} b^{2} c d \,e^{2} f +\frac {6}{7} x^{7} a b \,c^{2} e \,f^{2}+\frac {2}{3} x^{9} a b \,d^{2} e^{2} f +\frac {1}{3} x^{9} b^{2} c^{2} e \,f^{2}+\frac {4}{3} x^{9} a b c d e \,f^{2}+\frac {6}{11} x^{11} b^{2} c d e \,f^{2}+\frac {1}{11} x^{11} b^{2} c^{2} f^{3}+\frac {1}{9} x^{9} b^{2} d^{2} e^{3}+\frac {1}{11} x^{11} a^{2} d^{2} f^{3}+\frac {12}{7} x^{7} a b c d \,e^{2} f +\frac {6}{11} x^{11} a b \,d^{2} e \,f^{2}+\frac {4}{11} x^{11} a b c d \,f^{3}+a^{2} c^{2} e^{3} x +\frac {1}{15} b^{2} d^{2} f^{3} x^{15}\)	\(496\)
parallelrisch	\(\frac {6}{5} x^{5} a^{2} c d \,e^{2} f +x^{3} a^{2} c^{2} e^{2} f +\frac {2}{3} x^{3} a^{2} c d \,e^{3}+\frac {1}{5} x^{5} a^{2} d^{2} e^{3}+\frac {3}{5} x^{5} a^{2} c^{2} e \,f^{2}+\frac {3}{7} x^{7} a^{2} d^{2} e^{2} f +\frac {1}{5} x^{5} b^{2} c^{2} e^{3}+\frac {1}{7} x^{7} a^{2} c^{2} f^{3}+\frac {6}{5} x^{5} a b \,c^{2} e^{2} f +\frac {3}{11} x^{11} b^{2} d^{2} e^{2} f +\frac {2}{13} x^{13} a b \,d^{2} f^{3}+\frac {2}{13} x^{13} b^{2} c d \,f^{3}+\frac {3}{13} x^{13} b^{2} d^{2} e \,f^{2}+\frac {2}{7} x^{7} a b \,d^{2} e^{3}+\frac {3}{7} x^{7} b^{2} c^{2} e^{2} f +\frac {2}{7} x^{7} b^{2} d \,e^{3} c +\frac {2}{9} x^{9} a^{2} c d \,f^{3}+\frac {1}{3} x^{9} a^{2} d^{2} e \,f^{2}+\frac {2}{9} x^{9} a b \,c^{2} f^{3}+\frac {4}{5} x^{5} a b c d \,e^{3}+\frac {6}{7} x^{7} a^{2} c d e \,f^{2}+\frac {2}{3} x^{3} a b \,c^{2} e^{3}+\frac {2}{3} x^{9} b^{2} c d \,e^{2} f +\frac {6}{7} x^{7} a b \,c^{2} e \,f^{2}+\frac {2}{3} x^{9} a b \,d^{2} e^{2} f +\frac {1}{3} x^{9} b^{2} c^{2} e \,f^{2}+\frac {4}{3} x^{9} a b c d e \,f^{2}+\frac {6}{11} x^{11} b^{2} c d e \,f^{2}+\frac {1}{11} x^{11} b^{2} c^{2} f^{3}+\frac {1}{9} x^{9} b^{2} d^{2} e^{3}+\frac {1}{11} x^{11} a^{2} d^{2} f^{3}+\frac {12}{7} x^{7} a b c d \,e^{2} f +\frac {6}{11} x^{11} a b \,d^{2} e \,f^{2}+\frac {4}{11} x^{11} a b c d \,f^{3}+a^{2} c^{2} e^{3} x +\frac {1}{15} b^{2} d^{2} f^{3} x^{15}\)	\(496\)
orering	\(\frac {x \left (3003 b^{2} d^{2} f^{3} x^{14}+6930 a b \,d^{2} f^{3} x^{12}+6930 b^{2} c d \,f^{3} x^{12}+10395 b^{2} d^{2} e \,f^{2} x^{12}+4095 a^{2} d^{2} f^{3} x^{10}+16380 a b c d \,f^{3} x^{10}+24570 a b \,d^{2} e \,f^{2} x^{10}+4095 b^{2} c^{2} f^{3} x^{10}+24570 b^{2} c d e \,f^{2} x^{10}+12285 b^{2} d^{2} e^{2} f \,x^{10}+10010 a^{2} c d \,f^{3} x^{8}+15015 a^{2} d^{2} e \,f^{2} x^{8}+10010 a b \,c^{2} f^{3} x^{8}+60060 a b c d e \,f^{2} x^{8}+30030 a b \,d^{2} e^{2} f \,x^{8}+15015 b^{2} c^{2} e \,f^{2} x^{8}+30030 b^{2} c d \,e^{2} f \,x^{8}+5005 b^{2} d^{2} e^{3} x^{8}+6435 a^{2} c^{2} f^{3} x^{6}+38610 a^{2} c d e \,f^{2} x^{6}+19305 a^{2} d^{2} e^{2} f \,x^{6}+38610 a b \,c^{2} e \,f^{2} x^{6}+77220 a b c d \,e^{2} f \,x^{6}+12870 a b \,d^{2} e^{3} x^{6}+19305 b^{2} c^{2} e^{2} f \,x^{6}+12870 b^{2} c d \,e^{3} x^{6}+27027 a^{2} c^{2} e \,f^{2} x^{4}+54054 a^{2} c d \,e^{2} f \,x^{4}+9009 a^{2} d^{2} e^{3} x^{4}+54054 a b \,c^{2} e^{2} f \,x^{4}+36036 a b c d \,e^{3} x^{4}+9009 b^{2} c^{2} e^{3} x^{4}+45045 a^{2} c^{2} e^{2} f \,x^{2}+30030 a^{2} c d \,e^{3} x^{2}+30030 a b \,c^{2} e^{3} x^{2}+45045 a^{2} c^{2} e^{3}\right )}{45045}\)	\(500\)

Input:

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

Output:

1/15*b^2*d^2*f^3*x^15+1/13*((2*a*b*d^2+2*b^2*c*d)*f^3+3*b^2*d^2*e*f^2)*x^1 
3+1/11*((a^2*d^2+4*a*b*c*d+b^2*c^2)*f^3+3*(2*a*b*d^2+2*b^2*c*d)*e*f^2+3*b^ 
2*d^2*e^2*f)*x^11+1/9*((2*a^2*c*d+2*a*b*c^2)*f^3+3*(a^2*d^2+4*a*b*c*d+b^2* 
c^2)*e*f^2+3*(2*a*b*d^2+2*b^2*c*d)*e^2*f+b^2*d^2*e^3)*x^9+1/7*(a^2*c^2*f^3 
+3*(2*a^2*c*d+2*a*b*c^2)*e*f^2+3*(a^2*d^2+4*a*b*c*d+b^2*c^2)*e^2*f+(2*a*b* 
d^2+2*b^2*c*d)*e^3)*x^7+1/5*(3*a^2*c^2*e*f^2+3*(2*a^2*c*d+2*a*b*c^2)*e^2*f 
+(a^2*d^2+4*a*b*c*d+b^2*c^2)*e^3)*x^5+1/3*(3*a^2*c^2*e^2*f+(2*a^2*c*d+2*a* 
b*c^2)*e^3)*x^3+a^2*c^2*e^3*x
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\frac {1}{15} \, b^{2} d^{2} f^{3} x^{15} + \frac {1}{13} \, {\left (3 \, b^{2} d^{2} e f^{2} + 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{3}\right )} x^{13} + \frac {1}{11} \, {\left (3 \, b^{2} d^{2} e^{2} f + 6 \, {\left (b^{2} c d + a b d^{2}\right )} e f^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{3}\right )} x^{11} + \frac {1}{9} \, {\left (b^{2} d^{2} e^{3} + 6 \, {\left (b^{2} c d + a b d^{2}\right )} e^{2} f + 3 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e f^{2} + 2 \, {\left (a b c^{2} + a^{2} c d\right )} f^{3}\right )} x^{9} + a^{2} c^{2} e^{3} x + \frac {1}{7} \, {\left (a^{2} c^{2} f^{3} + 2 \, {\left (b^{2} c d + a b d^{2}\right )} e^{3} + 3 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e^{2} f + 6 \, {\left (a b c^{2} + a^{2} c d\right )} e f^{2}\right )} x^{7} + \frac {1}{5} \, {\left (3 \, a^{2} c^{2} e f^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e^{3} + 6 \, {\left (a b c^{2} + a^{2} c d\right )} e^{2} f\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a^{2} c^{2} e^{2} f + 2 \, {\left (a b c^{2} + a^{2} c d\right )} e^{3}\right )} x^{3} \] Input:

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

Output:

1/15*b^2*d^2*f^3*x^15 + 1/13*(3*b^2*d^2*e*f^2 + 2*(b^2*c*d + a*b*d^2)*f^3) 
*x^13 + 1/11*(3*b^2*d^2*e^2*f + 6*(b^2*c*d + a*b*d^2)*e*f^2 + (b^2*c^2 + 4 
*a*b*c*d + a^2*d^2)*f^3)*x^11 + 1/9*(b^2*d^2*e^3 + 6*(b^2*c*d + a*b*d^2)*e 
^2*f + 3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e*f^2 + 2*(a*b*c^2 + a^2*c*d)*f^3 
)*x^9 + a^2*c^2*e^3*x + 1/7*(a^2*c^2*f^3 + 2*(b^2*c*d + a*b*d^2)*e^3 + 3*( 
b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2*f + 6*(a*b*c^2 + a^2*c*d)*e*f^2)*x^7 + 
1/5*(3*a^2*c^2*e*f^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^3 + 6*(a*b*c^2 + 
a^2*c*d)*e^2*f)*x^5 + 1/3*(3*a^2*c^2*e^2*f + 2*(a*b*c^2 + a^2*c*d)*e^3)*x^ 
3
 

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.44 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=a^{2} c^{2} e^{3} x + \frac {b^{2} d^{2} f^{3} x^{15}}{15} + x^{13} \cdot \left (\frac {2 a b d^{2} f^{3}}{13} + \frac {2 b^{2} c d f^{3}}{13} + \frac {3 b^{2} d^{2} e f^{2}}{13}\right ) + x^{11} \left (\frac {a^{2} d^{2} f^{3}}{11} + \frac {4 a b c d f^{3}}{11} + \frac {6 a b d^{2} e f^{2}}{11} + \frac {b^{2} c^{2} f^{3}}{11} + \frac {6 b^{2} c d e f^{2}}{11} + \frac {3 b^{2} d^{2} e^{2} f}{11}\right ) + x^{9} \cdot \left (\frac {2 a^{2} c d f^{3}}{9} + \frac {a^{2} d^{2} e f^{2}}{3} + \frac {2 a b c^{2} f^{3}}{9} + \frac {4 a b c d e f^{2}}{3} + \frac {2 a b d^{2} e^{2} f}{3} + \frac {b^{2} c^{2} e f^{2}}{3} + \frac {2 b^{2} c d e^{2} f}{3} + \frac {b^{2} d^{2} e^{3}}{9}\right ) + x^{7} \left (\frac {a^{2} c^{2} f^{3}}{7} + \frac {6 a^{2} c d e f^{2}}{7} + \frac {3 a^{2} d^{2} e^{2} f}{7} + \frac {6 a b c^{2} e f^{2}}{7} + \frac {12 a b c d e^{2} f}{7} + \frac {2 a b d^{2} e^{3}}{7} + \frac {3 b^{2} c^{2} e^{2} f}{7} + \frac {2 b^{2} c d e^{3}}{7}\right ) + x^{5} \cdot \left (\frac {3 a^{2} c^{2} e f^{2}}{5} + \frac {6 a^{2} c d e^{2} f}{5} + \frac {a^{2} d^{2} e^{3}}{5} + \frac {6 a b c^{2} e^{2} f}{5} + \frac {4 a b c d e^{3}}{5} + \frac {b^{2} c^{2} e^{3}}{5}\right ) + x^{3} \left (a^{2} c^{2} e^{2} f + \frac {2 a^{2} c d e^{3}}{3} + \frac {2 a b c^{2} e^{3}}{3}\right ) \] Input:

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

Output:

a**2*c**2*e**3*x + b**2*d**2*f**3*x**15/15 + x**13*(2*a*b*d**2*f**3/13 + 2 
*b**2*c*d*f**3/13 + 3*b**2*d**2*e*f**2/13) + x**11*(a**2*d**2*f**3/11 + 4* 
a*b*c*d*f**3/11 + 6*a*b*d**2*e*f**2/11 + b**2*c**2*f**3/11 + 6*b**2*c*d*e* 
f**2/11 + 3*b**2*d**2*e**2*f/11) + x**9*(2*a**2*c*d*f**3/9 + a**2*d**2*e*f 
**2/3 + 2*a*b*c**2*f**3/9 + 4*a*b*c*d*e*f**2/3 + 2*a*b*d**2*e**2*f/3 + b** 
2*c**2*e*f**2/3 + 2*b**2*c*d*e**2*f/3 + b**2*d**2*e**3/9) + x**7*(a**2*c** 
2*f**3/7 + 6*a**2*c*d*e*f**2/7 + 3*a**2*d**2*e**2*f/7 + 6*a*b*c**2*e*f**2/ 
7 + 12*a*b*c*d*e**2*f/7 + 2*a*b*d**2*e**3/7 + 3*b**2*c**2*e**2*f/7 + 2*b** 
2*c*d*e**3/7) + x**5*(3*a**2*c**2*e*f**2/5 + 6*a**2*c*d*e**2*f/5 + a**2*d* 
*2*e**3/5 + 6*a*b*c**2*e**2*f/5 + 4*a*b*c*d*e**3/5 + b**2*c**2*e**3/5) + x 
**3*(a**2*c**2*e**2*f + 2*a**2*c*d*e**3/3 + 2*a*b*c**2*e**3/3)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\frac {1}{15} \, b^{2} d^{2} f^{3} x^{15} + \frac {1}{13} \, {\left (3 \, b^{2} d^{2} e f^{2} + 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{3}\right )} x^{13} + \frac {1}{11} \, {\left (3 \, b^{2} d^{2} e^{2} f + 6 \, {\left (b^{2} c d + a b d^{2}\right )} e f^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{3}\right )} x^{11} + \frac {1}{9} \, {\left (b^{2} d^{2} e^{3} + 6 \, {\left (b^{2} c d + a b d^{2}\right )} e^{2} f + 3 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e f^{2} + 2 \, {\left (a b c^{2} + a^{2} c d\right )} f^{3}\right )} x^{9} + a^{2} c^{2} e^{3} x + \frac {1}{7} \, {\left (a^{2} c^{2} f^{3} + 2 \, {\left (b^{2} c d + a b d^{2}\right )} e^{3} + 3 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e^{2} f + 6 \, {\left (a b c^{2} + a^{2} c d\right )} e f^{2}\right )} x^{7} + \frac {1}{5} \, {\left (3 \, a^{2} c^{2} e f^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e^{3} + 6 \, {\left (a b c^{2} + a^{2} c d\right )} e^{2} f\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a^{2} c^{2} e^{2} f + 2 \, {\left (a b c^{2} + a^{2} c d\right )} e^{3}\right )} x^{3} \] Input:

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

Output:

1/15*b^2*d^2*f^3*x^15 + 1/13*(3*b^2*d^2*e*f^2 + 2*(b^2*c*d + a*b*d^2)*f^3) 
*x^13 + 1/11*(3*b^2*d^2*e^2*f + 6*(b^2*c*d + a*b*d^2)*e*f^2 + (b^2*c^2 + 4 
*a*b*c*d + a^2*d^2)*f^3)*x^11 + 1/9*(b^2*d^2*e^3 + 6*(b^2*c*d + a*b*d^2)*e 
^2*f + 3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e*f^2 + 2*(a*b*c^2 + a^2*c*d)*f^3 
)*x^9 + a^2*c^2*e^3*x + 1/7*(a^2*c^2*f^3 + 2*(b^2*c*d + a*b*d^2)*e^3 + 3*( 
b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2*f + 6*(a*b*c^2 + a^2*c*d)*e*f^2)*x^7 + 
1/5*(3*a^2*c^2*e*f^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^3 + 6*(a*b*c^2 + 
a^2*c*d)*e^2*f)*x^5 + 1/3*(3*a^2*c^2*e^2*f + 2*(a*b*c^2 + a^2*c*d)*e^3)*x^ 
3
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.37 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\frac {1}{15} \, b^{2} d^{2} f^{3} x^{15} + \frac {3}{13} \, b^{2} d^{2} e f^{2} x^{13} + \frac {2}{13} \, b^{2} c d f^{3} x^{13} + \frac {2}{13} \, a b d^{2} f^{3} x^{13} + \frac {3}{11} \, b^{2} d^{2} e^{2} f x^{11} + \frac {6}{11} \, b^{2} c d e f^{2} x^{11} + \frac {6}{11} \, a b d^{2} e f^{2} x^{11} + \frac {1}{11} \, b^{2} c^{2} f^{3} x^{11} + \frac {4}{11} \, a b c d f^{3} x^{11} + \frac {1}{11} \, a^{2} d^{2} f^{3} x^{11} + \frac {1}{9} \, b^{2} d^{2} e^{3} x^{9} + \frac {2}{3} \, b^{2} c d e^{2} f x^{9} + \frac {2}{3} \, a b d^{2} e^{2} f x^{9} + \frac {1}{3} \, b^{2} c^{2} e f^{2} x^{9} + \frac {4}{3} \, a b c d e f^{2} x^{9} + \frac {1}{3} \, a^{2} d^{2} e f^{2} x^{9} + \frac {2}{9} \, a b c^{2} f^{3} x^{9} + \frac {2}{9} \, a^{2} c d f^{3} x^{9} + \frac {2}{7} \, b^{2} c d e^{3} x^{7} + \frac {2}{7} \, a b d^{2} e^{3} x^{7} + \frac {3}{7} \, b^{2} c^{2} e^{2} f x^{7} + \frac {12}{7} \, a b c d e^{2} f x^{7} + \frac {3}{7} \, a^{2} d^{2} e^{2} f x^{7} + \frac {6}{7} \, a b c^{2} e f^{2} x^{7} + \frac {6}{7} \, a^{2} c d e f^{2} x^{7} + \frac {1}{7} \, a^{2} c^{2} f^{3} x^{7} + \frac {1}{5} \, b^{2} c^{2} e^{3} x^{5} + \frac {4}{5} \, a b c d e^{3} x^{5} + \frac {1}{5} \, a^{2} d^{2} e^{3} x^{5} + \frac {6}{5} \, a b c^{2} e^{2} f x^{5} + \frac {6}{5} \, a^{2} c d e^{2} f x^{5} + \frac {3}{5} \, a^{2} c^{2} e f^{2} x^{5} + \frac {2}{3} \, a b c^{2} e^{3} x^{3} + \frac {2}{3} \, a^{2} c d e^{3} x^{3} + a^{2} c^{2} e^{2} f x^{3} + a^{2} c^{2} e^{3} x \] Input:

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

Output:

1/15*b^2*d^2*f^3*x^15 + 3/13*b^2*d^2*e*f^2*x^13 + 2/13*b^2*c*d*f^3*x^13 + 
2/13*a*b*d^2*f^3*x^13 + 3/11*b^2*d^2*e^2*f*x^11 + 6/11*b^2*c*d*e*f^2*x^11 
+ 6/11*a*b*d^2*e*f^2*x^11 + 1/11*b^2*c^2*f^3*x^11 + 4/11*a*b*c*d*f^3*x^11 
+ 1/11*a^2*d^2*f^3*x^11 + 1/9*b^2*d^2*e^3*x^9 + 2/3*b^2*c*d*e^2*f*x^9 + 2/ 
3*a*b*d^2*e^2*f*x^9 + 1/3*b^2*c^2*e*f^2*x^9 + 4/3*a*b*c*d*e*f^2*x^9 + 1/3* 
a^2*d^2*e*f^2*x^9 + 2/9*a*b*c^2*f^3*x^9 + 2/9*a^2*c*d*f^3*x^9 + 2/7*b^2*c* 
d*e^3*x^7 + 2/7*a*b*d^2*e^3*x^7 + 3/7*b^2*c^2*e^2*f*x^7 + 12/7*a*b*c*d*e^2 
*f*x^7 + 3/7*a^2*d^2*e^2*f*x^7 + 6/7*a*b*c^2*e*f^2*x^7 + 6/7*a^2*c*d*e*f^2 
*x^7 + 1/7*a^2*c^2*f^3*x^7 + 1/5*b^2*c^2*e^3*x^5 + 4/5*a*b*c*d*e^3*x^5 + 1 
/5*a^2*d^2*e^3*x^5 + 6/5*a*b*c^2*e^2*f*x^5 + 6/5*a^2*c*d*e^2*f*x^5 + 3/5*a 
^2*c^2*e*f^2*x^5 + 2/3*a*b*c^2*e^3*x^3 + 2/3*a^2*c*d*e^3*x^3 + a^2*c^2*e^2 
*f*x^3 + a^2*c^2*e^3*x
 

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.11 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=x^5\,\left (\frac {3\,a^2\,c^2\,e\,f^2}{5}+\frac {6\,a^2\,c\,d\,e^2\,f}{5}+\frac {a^2\,d^2\,e^3}{5}+\frac {6\,a\,b\,c^2\,e^2\,f}{5}+\frac {4\,a\,b\,c\,d\,e^3}{5}+\frac {b^2\,c^2\,e^3}{5}\right )+x^{11}\,\left (\frac {a^2\,d^2\,f^3}{11}+\frac {4\,a\,b\,c\,d\,f^3}{11}+\frac {6\,a\,b\,d^2\,e\,f^2}{11}+\frac {b^2\,c^2\,f^3}{11}+\frac {6\,b^2\,c\,d\,e\,f^2}{11}+\frac {3\,b^2\,d^2\,e^2\,f}{11}\right )+x^7\,\left (\frac {a^2\,c^2\,f^3}{7}+\frac {6\,a^2\,c\,d\,e\,f^2}{7}+\frac {3\,a^2\,d^2\,e^2\,f}{7}+\frac {6\,a\,b\,c^2\,e\,f^2}{7}+\frac {12\,a\,b\,c\,d\,e^2\,f}{7}+\frac {2\,a\,b\,d^2\,e^3}{7}+\frac {3\,b^2\,c^2\,e^2\,f}{7}+\frac {2\,b^2\,c\,d\,e^3}{7}\right )+x^9\,\left (\frac {2\,a^2\,c\,d\,f^3}{9}+\frac {a^2\,d^2\,e\,f^2}{3}+\frac {2\,a\,b\,c^2\,f^3}{9}+\frac {4\,a\,b\,c\,d\,e\,f^2}{3}+\frac {2\,a\,b\,d^2\,e^2\,f}{3}+\frac {b^2\,c^2\,e\,f^2}{3}+\frac {2\,b^2\,c\,d\,e^2\,f}{3}+\frac {b^2\,d^2\,e^3}{9}\right )+a^2\,c^2\,e^3\,x+\frac {b^2\,d^2\,f^3\,x^{15}}{15}+\frac {a\,c\,e^2\,x^3\,\left (3\,a\,c\,f+2\,a\,d\,e+2\,b\,c\,e\right )}{3}+\frac {b\,d\,f^2\,x^{13}\,\left (2\,a\,d\,f+2\,b\,c\,f+3\,b\,d\,e\right )}{13} \] Input:

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

Output:

x^5*((a^2*d^2*e^3)/5 + (b^2*c^2*e^3)/5 + (3*a^2*c^2*e*f^2)/5 + (6*a*b*c^2* 
e^2*f)/5 + (6*a^2*c*d*e^2*f)/5 + (4*a*b*c*d*e^3)/5) + x^11*((a^2*d^2*f^3)/ 
11 + (b^2*c^2*f^3)/11 + (3*b^2*d^2*e^2*f)/11 + (6*a*b*d^2*e*f^2)/11 + (6*b 
^2*c*d*e*f^2)/11 + (4*a*b*c*d*f^3)/11) + x^7*((a^2*c^2*f^3)/7 + (3*a^2*d^2 
*e^2*f)/7 + (3*b^2*c^2*e^2*f)/7 + (2*a*b*d^2*e^3)/7 + (2*b^2*c*d*e^3)/7 + 
(6*a*b*c^2*e*f^2)/7 + (6*a^2*c*d*e*f^2)/7 + (12*a*b*c*d*e^2*f)/7) + x^9*(( 
b^2*d^2*e^3)/9 + (a^2*d^2*e*f^2)/3 + (b^2*c^2*e*f^2)/3 + (2*a*b*c^2*f^3)/9 
 + (2*a^2*c*d*f^3)/9 + (2*a*b*d^2*e^2*f)/3 + (2*b^2*c*d*e^2*f)/3 + (4*a*b* 
c*d*e*f^2)/3) + a^2*c^2*e^3*x + (b^2*d^2*f^3*x^15)/15 + (a*c*e^2*x^3*(3*a* 
c*f + 2*a*d*e + 2*b*c*e))/3 + (b*d*f^2*x^13*(2*a*d*f + 2*b*c*f + 3*b*d*e)) 
/13
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.38 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\frac {x \left (3003 b^{2} d^{2} f^{3} x^{14}+6930 a b \,d^{2} f^{3} x^{12}+6930 b^{2} c d \,f^{3} x^{12}+10395 b^{2} d^{2} e \,f^{2} x^{12}+4095 a^{2} d^{2} f^{3} x^{10}+16380 a b c d \,f^{3} x^{10}+24570 a b \,d^{2} e \,f^{2} x^{10}+4095 b^{2} c^{2} f^{3} x^{10}+24570 b^{2} c d e \,f^{2} x^{10}+12285 b^{2} d^{2} e^{2} f \,x^{10}+10010 a^{2} c d \,f^{3} x^{8}+15015 a^{2} d^{2} e \,f^{2} x^{8}+10010 a b \,c^{2} f^{3} x^{8}+60060 a b c d e \,f^{2} x^{8}+30030 a b \,d^{2} e^{2} f \,x^{8}+15015 b^{2} c^{2} e \,f^{2} x^{8}+30030 b^{2} c d \,e^{2} f \,x^{8}+5005 b^{2} d^{2} e^{3} x^{8}+6435 a^{2} c^{2} f^{3} x^{6}+38610 a^{2} c d e \,f^{2} x^{6}+19305 a^{2} d^{2} e^{2} f \,x^{6}+38610 a b \,c^{2} e \,f^{2} x^{6}+77220 a b c d \,e^{2} f \,x^{6}+12870 a b \,d^{2} e^{3} x^{6}+19305 b^{2} c^{2} e^{2} f \,x^{6}+12870 b^{2} c d \,e^{3} x^{6}+27027 a^{2} c^{2} e \,f^{2} x^{4}+54054 a^{2} c d \,e^{2} f \,x^{4}+9009 a^{2} d^{2} e^{3} x^{4}+54054 a b \,c^{2} e^{2} f \,x^{4}+36036 a b c d \,e^{3} x^{4}+9009 b^{2} c^{2} e^{3} x^{4}+45045 a^{2} c^{2} e^{2} f \,x^{2}+30030 a^{2} c d \,e^{3} x^{2}+30030 a b \,c^{2} e^{3} x^{2}+45045 a^{2} c^{2} e^{3}\right )}{45045} \] Input:

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

Output:

(x*(45045*a**2*c**2*e**3 + 45045*a**2*c**2*e**2*f*x**2 + 27027*a**2*c**2*e 
*f**2*x**4 + 6435*a**2*c**2*f**3*x**6 + 30030*a**2*c*d*e**3*x**2 + 54054*a 
**2*c*d*e**2*f*x**4 + 38610*a**2*c*d*e*f**2*x**6 + 10010*a**2*c*d*f**3*x** 
8 + 9009*a**2*d**2*e**3*x**4 + 19305*a**2*d**2*e**2*f*x**6 + 15015*a**2*d* 
*2*e*f**2*x**8 + 4095*a**2*d**2*f**3*x**10 + 30030*a*b*c**2*e**3*x**2 + 54 
054*a*b*c**2*e**2*f*x**4 + 38610*a*b*c**2*e*f**2*x**6 + 10010*a*b*c**2*f** 
3*x**8 + 36036*a*b*c*d*e**3*x**4 + 77220*a*b*c*d*e**2*f*x**6 + 60060*a*b*c 
*d*e*f**2*x**8 + 16380*a*b*c*d*f**3*x**10 + 12870*a*b*d**2*e**3*x**6 + 300 
30*a*b*d**2*e**2*f*x**8 + 24570*a*b*d**2*e*f**2*x**10 + 6930*a*b*d**2*f**3 
*x**12 + 9009*b**2*c**2*e**3*x**4 + 19305*b**2*c**2*e**2*f*x**6 + 15015*b* 
*2*c**2*e*f**2*x**8 + 4095*b**2*c**2*f**3*x**10 + 12870*b**2*c*d*e**3*x**6 
 + 30030*b**2*c*d*e**2*f*x**8 + 24570*b**2*c*d*e*f**2*x**10 + 6930*b**2*c* 
d*f**3*x**12 + 5005*b**2*d**2*e**3*x**8 + 12285*b**2*d**2*e**2*f*x**10 + 1 
0395*b**2*d**2*e*f**2*x**12 + 3003*b**2*d**2*f**3*x**14))/45045