\(\int (a+b x^2) (c+d x^2)^3 (e+f x^2)^3 \, dx\) [214]

Optimal result

Integrand size = 26, antiderivative size = 310 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=a c^3 e^3 x+\frac {1}{3} c^2 e^2 (b c e+3 a (d e+c f)) x^3+\frac {3}{5} c e \left (b c e (d e+c f)+a \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (3 b c e \left (d^2 e^2+3 c d e f+c^2 f^2\right )+a \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^7+\frac {1}{9} \left (3 a d f \left (d^2 e^2+3 c d e f+c^2 f^2\right )+b \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^9+\frac {3}{11} d f \left (a d f (d e+c f)+b \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^{11}+\frac {1}{13} d^2 f^2 (a d f+3 b (d e+c f)) x^{13}+\frac {1}{15} b d^3 f^3 x^{15} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=a c^3 e^3 x+\frac {1}{3} c^2 e^2 (b c e+3 a (d e+c f)) x^3+\frac {3}{5} c e \left (b c e (d e+c f)+a \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (3 b c e \left (d^2 e^2+3 c d e f+c^2 f^2\right )+a \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^7+\frac {1}{9} \left (3 a d f \left (d^2 e^2+3 c d e f+c^2 f^2\right )+b \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^9+\frac {3}{11} d f \left (a d f (d e+c f)+b \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^{11}+\frac {1}{13} d^2 f^2 (a d f+3 b (d e+c f)) x^{13}+\frac {1}{15} b d^3 f^3 x^{15} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.

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

\(\Big \downarrow \) 396

\(\displaystyle \int \left (3 d f x^{10} \left (a d f (c f+d e)+b \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+3 c e x^4 \left (a \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b c e (c f+d e)\right )+c^2 e^2 x^2 (3 a (c f+d e)+b c e)+x^8 \left (3 a d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )\right )+x^6 \left (a \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )+3 b c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+d^2 f^2 x^{12} (a d f+3 b (c f+d e))+a c^3 e^3+b d^3 f^3 x^{14}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

a*c^3*e^3*x + (c^2*e^2*(b*c*e + 3*a*(d*e + c*f))*x^3)/3 + (3*c*e*(b*c*e*(d 
*e + c*f) + a*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x^5)/5 + ((3*b*c*e*(d^2*e^2 
 + 3*c*d*e*f + c^2*f^2) + a*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3 
*f^3))*x^7)/7 + ((3*a*d*f*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + b*(d^3*e^3 + 9 
*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3*f^3))*x^9)/9 + (3*d*f*(a*d*f*(d*e + c*f 
) + b*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x^11)/11 + (d^2*f^2*(a*d*f + 3*b*(d 
*e + c*f))*x^13)/13 + (b*d^3*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 = 0.51 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.09

method	result	size
default	\(\frac {b \,d^{3} f^{3} x^{15}}{15}+\frac {\left (\left (a \,d^{3}+3 b c \,d^{2}\right ) f^{3}+3 b \,d^{3} e \,f^{2}\right ) x^{13}}{13}+\frac {\left (\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) f^{3}+3 \left (a \,d^{3}+3 b c \,d^{2}\right ) e \,f^{2}+3 b \,d^{3} e^{2} f \right ) x^{11}}{11}+\frac {\left (\left (3 a \,c^{2} d +c^{3} b \right ) f^{3}+3 \left (3 a c \,d^{2}+3 b \,c^{2} d \right ) e \,f^{2}+3 \left (a \,d^{3}+3 b c \,d^{2}\right ) e^{2} f +b \,d^{3} e^{3}\right ) x^{9}}{9}+\frac {\left (c^{3} a \,f^{3}+3 \left (3 a \,c^{2} d +c^{3} b \right ) e \,f^{2}+3 \left (3 a c \,d^{2}+3 b \,c^{2} d \right ) e^{2} f +\left (a \,d^{3}+3 b c \,d^{2}\right ) e^{3}\right ) x^{7}}{7}+\frac {\left (3 c^{3} a e \,f^{2}+3 \left (3 a \,c^{2} d +c^{3} b \right ) e^{2} f +\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) e^{3}\right ) x^{5}}{5}+\frac {\left (3 c^{3} a \,e^{2} f +\left (3 a \,c^{2} d +c^{3} b \right ) e^{3}\right ) x^{3}}{3}+a \,c^{3} e^{3} x\)	\(339\)
norman	\(a \,c^{3} e^{3} x +\left (c^{3} a \,e^{2} f +a \,c^{2} d \,e^{3}+\frac {1}{3} b \,c^{3} e^{3}\right ) x^{3}+\left (\frac {3}{5} c^{3} a e \,f^{2}+\frac {9}{5} a \,c^{2} d \,e^{2} f +\frac {3}{5} a c \,d^{2} e^{3}+\frac {3}{5} b \,c^{3} e^{2} f +\frac {3}{5} b \,c^{2} d \,e^{3}\right ) x^{5}+\left (\frac {1}{7} c^{3} a \,f^{3}+\frac {9}{7} a \,c^{2} d e \,f^{2}+\frac {9}{7} a c \,d^{2} e^{2} f +\frac {1}{7} a \,d^{3} e^{3}+\frac {3}{7} b \,c^{3} e \,f^{2}+\frac {9}{7} b \,c^{2} d \,e^{2} f +\frac {3}{7} b c \,d^{2} e^{3}\right ) x^{7}+\left (\frac {1}{3} a \,c^{2} d \,f^{3}+a c \,d^{2} e \,f^{2}+\frac {1}{3} a \,d^{3} e^{2} f +\frac {1}{9} b \,c^{3} f^{3}+b \,c^{2} d e \,f^{2}+b c \,d^{2} e^{2} f +\frac {1}{9} b \,d^{3} e^{3}\right ) x^{9}+\left (\frac {3}{11} a c \,d^{2} f^{3}+\frac {3}{11} a \,d^{3} e \,f^{2}+\frac {3}{11} b \,c^{2} d \,f^{3}+\frac {9}{11} b c \,d^{2} e \,f^{2}+\frac {3}{11} b \,d^{3} e^{2} f \right ) x^{11}+\left (\frac {1}{13} a \,d^{3} f^{3}+\frac {3}{13} b c \,d^{2} f^{3}+\frac {3}{13} b \,d^{3} e \,f^{2}\right ) x^{13}+\frac {b \,d^{3} f^{3} x^{15}}{15}\)	\(350\)
gosper	\(\frac {9}{5} x^{5} a \,c^{2} d \,e^{2} f +x^{3} c^{3} a \,e^{2} f +x^{9} b c \,d^{2} e^{2} f +\frac {9}{7} x^{7} b \,c^{2} d \,e^{2} f +x^{9} a c \,d^{2} e \,f^{2}+x^{9} b \,c^{2} d e \,f^{2}+\frac {9}{11} x^{11} b c \,d^{2} e \,f^{2}+\frac {9}{7} x^{7} a c \,d^{2} e^{2} f +\frac {9}{7} x^{7} a \,c^{2} d e \,f^{2}+\frac {1}{9} x^{9} b \,d^{3} e^{3}+\frac {3}{11} x^{11} a \,d^{3} e \,f^{2}+\frac {3}{11} x^{11} b \,c^{2} d \,f^{3}+\frac {3}{11} x^{11} b \,d^{3} e^{2} f +\frac {3}{13} x^{13} b c \,d^{2} f^{3}+\frac {3}{13} x^{13} b \,d^{3} e \,f^{2}+\frac {1}{3} x^{3} b \,c^{3} e^{3}+\frac {1}{7} x^{7} c^{3} a \,f^{3}+\frac {1}{7} x^{7} a \,d^{3} e^{3}+\frac {1}{9} x^{9} b \,c^{3} f^{3}+\frac {3}{7} x^{7} b c \,d^{2} e^{3}+\frac {1}{3} x^{9} a \,c^{2} d \,f^{3}+\frac {1}{3} x^{9} a \,d^{3} e^{2} f +\frac {3}{11} x^{11} a c \,d^{2} f^{3}+x^{3} a \,c^{2} d \,e^{3}+\frac {3}{5} x^{5} c^{3} a e \,f^{2}+\frac {3}{5} x^{5} a c \,d^{2} e^{3}+\frac {3}{5} x^{5} b \,c^{3} e^{2} f +\frac {3}{5} x^{5} b \,c^{2} d \,e^{3}+\frac {3}{7} x^{7} b \,c^{3} e \,f^{2}+\frac {1}{13} x^{13} a \,d^{3} f^{3}+a \,c^{3} e^{3} x +\frac {1}{15} b \,d^{3} f^{3} x^{15}\)	\(410\)
risch	\(\frac {9}{5} x^{5} a \,c^{2} d \,e^{2} f +x^{3} c^{3} a \,e^{2} f +x^{9} b c \,d^{2} e^{2} f +\frac {9}{7} x^{7} b \,c^{2} d \,e^{2} f +x^{9} a c \,d^{2} e \,f^{2}+x^{9} b \,c^{2} d e \,f^{2}+\frac {9}{11} x^{11} b c \,d^{2} e \,f^{2}+\frac {9}{7} x^{7} a c \,d^{2} e^{2} f +\frac {9}{7} x^{7} a \,c^{2} d e \,f^{2}+\frac {1}{9} x^{9} b \,d^{3} e^{3}+\frac {3}{11} x^{11} a \,d^{3} e \,f^{2}+\frac {3}{11} x^{11} b \,c^{2} d \,f^{3}+\frac {3}{11} x^{11} b \,d^{3} e^{2} f +\frac {3}{13} x^{13} b c \,d^{2} f^{3}+\frac {3}{13} x^{13} b \,d^{3} e \,f^{2}+\frac {1}{3} x^{3} b \,c^{3} e^{3}+\frac {1}{7} x^{7} c^{3} a \,f^{3}+\frac {1}{7} x^{7} a \,d^{3} e^{3}+\frac {1}{9} x^{9} b \,c^{3} f^{3}+\frac {3}{7} x^{7} b c \,d^{2} e^{3}+\frac {1}{3} x^{9} a \,c^{2} d \,f^{3}+\frac {1}{3} x^{9} a \,d^{3} e^{2} f +\frac {3}{11} x^{11} a c \,d^{2} f^{3}+x^{3} a \,c^{2} d \,e^{3}+\frac {3}{5} x^{5} c^{3} a e \,f^{2}+\frac {3}{5} x^{5} a c \,d^{2} e^{3}+\frac {3}{5} x^{5} b \,c^{3} e^{2} f +\frac {3}{5} x^{5} b \,c^{2} d \,e^{3}+\frac {3}{7} x^{7} b \,c^{3} e \,f^{2}+\frac {1}{13} x^{13} a \,d^{3} f^{3}+a \,c^{3} e^{3} x +\frac {1}{15} b \,d^{3} f^{3} x^{15}\)	\(410\)
parallelrisch	\(\frac {9}{5} x^{5} a \,c^{2} d \,e^{2} f +x^{3} c^{3} a \,e^{2} f +x^{9} b c \,d^{2} e^{2} f +\frac {9}{7} x^{7} b \,c^{2} d \,e^{2} f +x^{9} a c \,d^{2} e \,f^{2}+x^{9} b \,c^{2} d e \,f^{2}+\frac {9}{11} x^{11} b c \,d^{2} e \,f^{2}+\frac {9}{7} x^{7} a c \,d^{2} e^{2} f +\frac {9}{7} x^{7} a \,c^{2} d e \,f^{2}+\frac {1}{9} x^{9} b \,d^{3} e^{3}+\frac {3}{11} x^{11} a \,d^{3} e \,f^{2}+\frac {3}{11} x^{11} b \,c^{2} d \,f^{3}+\frac {3}{11} x^{11} b \,d^{3} e^{2} f +\frac {3}{13} x^{13} b c \,d^{2} f^{3}+\frac {3}{13} x^{13} b \,d^{3} e \,f^{2}+\frac {1}{3} x^{3} b \,c^{3} e^{3}+\frac {1}{7} x^{7} c^{3} a \,f^{3}+\frac {1}{7} x^{7} a \,d^{3} e^{3}+\frac {1}{9} x^{9} b \,c^{3} f^{3}+\frac {3}{7} x^{7} b c \,d^{2} e^{3}+\frac {1}{3} x^{9} a \,c^{2} d \,f^{3}+\frac {1}{3} x^{9} a \,d^{3} e^{2} f +\frac {3}{11} x^{11} a c \,d^{2} f^{3}+x^{3} a \,c^{2} d \,e^{3}+\frac {3}{5} x^{5} c^{3} a e \,f^{2}+\frac {3}{5} x^{5} a c \,d^{2} e^{3}+\frac {3}{5} x^{5} b \,c^{3} e^{2} f +\frac {3}{5} x^{5} b \,c^{2} d \,e^{3}+\frac {3}{7} x^{7} b \,c^{3} e \,f^{2}+\frac {1}{13} x^{13} a \,d^{3} f^{3}+a \,c^{3} e^{3} x +\frac {1}{15} b \,d^{3} f^{3} x^{15}\)	\(410\)
orering	\(\frac {x \left (3003 b \,d^{3} f^{3} x^{14}+3465 a \,d^{3} f^{3} x^{12}+10395 b c \,d^{2} f^{3} x^{12}+10395 b \,d^{3} e \,f^{2} x^{12}+12285 a c \,d^{2} f^{3} x^{10}+12285 a \,d^{3} e \,f^{2} x^{10}+12285 b \,c^{2} d \,f^{3} x^{10}+36855 b c \,d^{2} e \,f^{2} x^{10}+12285 b \,d^{3} e^{2} f \,x^{10}+15015 a \,c^{2} d \,f^{3} x^{8}+45045 a c \,d^{2} e \,f^{2} x^{8}+15015 a \,d^{3} e^{2} f \,x^{8}+5005 b \,c^{3} f^{3} x^{8}+45045 b \,c^{2} d e \,f^{2} x^{8}+45045 b c \,d^{2} e^{2} f \,x^{8}+5005 b \,d^{3} e^{3} x^{8}+6435 a \,c^{3} f^{3} x^{6}+57915 a \,c^{2} d e \,f^{2} x^{6}+57915 a c \,d^{2} e^{2} f \,x^{6}+6435 a \,d^{3} e^{3} x^{6}+19305 b \,c^{3} e \,f^{2} x^{6}+57915 b \,c^{2} d \,e^{2} f \,x^{6}+19305 b c \,d^{2} e^{3} x^{6}+27027 a \,c^{3} e \,f^{2} x^{4}+81081 a \,c^{2} d \,e^{2} f \,x^{4}+27027 a c \,d^{2} e^{3} x^{4}+27027 b \,c^{3} e^{2} f \,x^{4}+27027 b \,c^{2} d \,e^{3} x^{4}+45045 a \,c^{3} e^{2} f \,x^{2}+45045 a \,c^{2} d \,e^{3} x^{2}+15015 b \,c^{3} e^{3} x^{2}+45045 c^{3} a \,e^{3}\right )}{45045}\)	\(418\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.05 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=\frac {1}{15} \, b d^{3} f^{3} x^{15} + \frac {1}{13} \, {\left (3 \, b d^{3} e f^{2} + {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3}\right )} x^{13} + \frac {3}{11} \, {\left (b d^{3} e^{2} f + {\left (3 \, b c d^{2} + a d^{3}\right )} e f^{2} + {\left (b c^{2} d + a c d^{2}\right )} f^{3}\right )} x^{11} + \frac {1}{9} \, {\left (b d^{3} e^{3} + 3 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e^{2} f + 9 \, {\left (b c^{2} d + a c d^{2}\right )} e f^{2} + {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{3}\right )} x^{9} + a c^{3} e^{3} x + \frac {1}{7} \, {\left (a c^{3} f^{3} + {\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} + 9 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} f + 3 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{2}\right )} x^{7} + \frac {3}{5} \, {\left (a c^{3} e f^{2} + {\left (b c^{2} d + a c d^{2}\right )} e^{3} + {\left (b c^{3} + 3 \, a c^{2} d\right )} e^{2} f\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a c^{3} e^{2} f + {\left (b c^{3} + 3 \, a c^{2} d\right )} e^{3}\right )} x^{3} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.36 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=a c^{3} e^{3} x + \frac {b d^{3} f^{3} x^{15}}{15} + x^{13} \left (\frac {a d^{3} f^{3}}{13} + \frac {3 b c d^{2} f^{3}}{13} + \frac {3 b d^{3} e f^{2}}{13}\right ) + x^{11} \cdot \left (\frac {3 a c d^{2} f^{3}}{11} + \frac {3 a d^{3} e f^{2}}{11} + \frac {3 b c^{2} d f^{3}}{11} + \frac {9 b c d^{2} e f^{2}}{11} + \frac {3 b d^{3} e^{2} f}{11}\right ) + x^{9} \left (\frac {a c^{2} d f^{3}}{3} + a c d^{2} e f^{2} + \frac {a d^{3} e^{2} f}{3} + \frac {b c^{3} f^{3}}{9} + b c^{2} d e f^{2} + b c d^{2} e^{2} f + \frac {b d^{3} e^{3}}{9}\right ) + x^{7} \left (\frac {a c^{3} f^{3}}{7} + \frac {9 a c^{2} d e f^{2}}{7} + \frac {9 a c d^{2} e^{2} f}{7} + \frac {a d^{3} e^{3}}{7} + \frac {3 b c^{3} e f^{2}}{7} + \frac {9 b c^{2} d e^{2} f}{7} + \frac {3 b c d^{2} e^{3}}{7}\right ) + x^{5} \cdot \left (\frac {3 a c^{3} e f^{2}}{5} + \frac {9 a c^{2} d e^{2} f}{5} + \frac {3 a c d^{2} e^{3}}{5} + \frac {3 b c^{3} e^{2} f}{5} + \frac {3 b c^{2} d e^{3}}{5}\right ) + x^{3} \left (a c^{3} e^{2} f + a c^{2} d e^{3} + \frac {b c^{3} e^{3}}{3}\right ) \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.05 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=\frac {1}{15} \, b d^{3} f^{3} x^{15} + \frac {1}{13} \, {\left (3 \, b d^{3} e f^{2} + {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3}\right )} x^{13} + \frac {3}{11} \, {\left (b d^{3} e^{2} f + {\left (3 \, b c d^{2} + a d^{3}\right )} e f^{2} + {\left (b c^{2} d + a c d^{2}\right )} f^{3}\right )} x^{11} + \frac {1}{9} \, {\left (b d^{3} e^{3} + 3 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e^{2} f + 9 \, {\left (b c^{2} d + a c d^{2}\right )} e f^{2} + {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{3}\right )} x^{9} + a c^{3} e^{3} x + \frac {1}{7} \, {\left (a c^{3} f^{3} + {\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} + 9 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} f + 3 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{2}\right )} x^{7} + \frac {3}{5} \, {\left (a c^{3} e f^{2} + {\left (b c^{2} d + a c d^{2}\right )} e^{3} + {\left (b c^{3} + 3 \, a c^{2} d\right )} e^{2} f\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a c^{3} e^{2} f + {\left (b c^{3} + 3 \, a c^{2} d\right )} e^{3}\right )} x^{3} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.32 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=\frac {1}{15} \, b d^{3} f^{3} x^{15} + \frac {3}{13} \, b d^{3} e f^{2} x^{13} + \frac {3}{13} \, b c d^{2} f^{3} x^{13} + \frac {1}{13} \, a d^{3} f^{3} x^{13} + \frac {3}{11} \, b d^{3} e^{2} f x^{11} + \frac {9}{11} \, b c d^{2} e f^{2} x^{11} + \frac {3}{11} \, a d^{3} e f^{2} x^{11} + \frac {3}{11} \, b c^{2} d f^{3} x^{11} + \frac {3}{11} \, a c d^{2} f^{3} x^{11} + \frac {1}{9} \, b d^{3} e^{3} x^{9} + b c d^{2} e^{2} f x^{9} + \frac {1}{3} \, a d^{3} e^{2} f x^{9} + b c^{2} d e f^{2} x^{9} + a c d^{2} e f^{2} x^{9} + \frac {1}{9} \, b c^{3} f^{3} x^{9} + \frac {1}{3} \, a c^{2} d f^{3} x^{9} + \frac {3}{7} \, b c d^{2} e^{3} x^{7} + \frac {1}{7} \, a d^{3} e^{3} x^{7} + \frac {9}{7} \, b c^{2} d e^{2} f x^{7} + \frac {9}{7} \, a c d^{2} e^{2} f x^{7} + \frac {3}{7} \, b c^{3} e f^{2} x^{7} + \frac {9}{7} \, a c^{2} d e f^{2} x^{7} + \frac {1}{7} \, a c^{3} f^{3} x^{7} + \frac {3}{5} \, b c^{2} d e^{3} x^{5} + \frac {3}{5} \, a c d^{2} e^{3} x^{5} + \frac {3}{5} \, b c^{3} e^{2} f x^{5} + \frac {9}{5} \, a c^{2} d e^{2} f x^{5} + \frac {3}{5} \, a c^{3} e f^{2} x^{5} + \frac {1}{3} \, b c^{3} e^{3} x^{3} + a c^{2} d e^{3} x^{3} + a c^{3} e^{2} f x^{3} + a c^{3} e^{3} x \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 1.74 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.08 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=x^5\,\left (\frac {3\,b\,c^3\,e^2\,f}{5}+\frac {3\,a\,c^3\,e\,f^2}{5}+\frac {3\,b\,c^2\,d\,e^3}{5}+\frac {9\,a\,c^2\,d\,e^2\,f}{5}+\frac {3\,a\,c\,d^2\,e^3}{5}\right )+x^{11}\,\left (\frac {3\,b\,c^2\,d\,f^3}{11}+\frac {9\,b\,c\,d^2\,e\,f^2}{11}+\frac {3\,a\,c\,d^2\,f^3}{11}+\frac {3\,b\,d^3\,e^2\,f}{11}+\frac {3\,a\,d^3\,e\,f^2}{11}\right )+x^7\,\left (\frac {3\,b\,c^3\,e\,f^2}{7}+\frac {a\,c^3\,f^3}{7}+\frac {9\,b\,c^2\,d\,e^2\,f}{7}+\frac {9\,a\,c^2\,d\,e\,f^2}{7}+\frac {3\,b\,c\,d^2\,e^3}{7}+\frac {9\,a\,c\,d^2\,e^2\,f}{7}+\frac {a\,d^3\,e^3}{7}\right )+x^9\,\left (\frac {b\,c^3\,f^3}{9}+b\,c^2\,d\,e\,f^2+\frac {a\,c^2\,d\,f^3}{3}+b\,c\,d^2\,e^2\,f+a\,c\,d^2\,e\,f^2+\frac {b\,d^3\,e^3}{9}+\frac {a\,d^3\,e^2\,f}{3}\right )+\frac {b\,d^3\,f^3\,x^{15}}{15}+\frac {c^2\,e^2\,x^3\,\left (3\,a\,c\,f+3\,a\,d\,e+b\,c\,e\right )}{3}+\frac {d^2\,f^2\,x^{13}\,\left (a\,d\,f+3\,b\,c\,f+3\,b\,d\,e\right )}{13}+a\,c^3\,e^3\,x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.35 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=\frac {x \left (3003 b \,d^{3} f^{3} x^{14}+3465 a \,d^{3} f^{3} x^{12}+10395 b c \,d^{2} f^{3} x^{12}+10395 b \,d^{3} e \,f^{2} x^{12}+12285 a c \,d^{2} f^{3} x^{10}+12285 a \,d^{3} e \,f^{2} x^{10}+12285 b \,c^{2} d \,f^{3} x^{10}+36855 b c \,d^{2} e \,f^{2} x^{10}+12285 b \,d^{3} e^{2} f \,x^{10}+15015 a \,c^{2} d \,f^{3} x^{8}+45045 a c \,d^{2} e \,f^{2} x^{8}+15015 a \,d^{3} e^{2} f \,x^{8}+5005 b \,c^{3} f^{3} x^{8}+45045 b \,c^{2} d e \,f^{2} x^{8}+45045 b c \,d^{2} e^{2} f \,x^{8}+5005 b \,d^{3} e^{3} x^{8}+6435 a \,c^{3} f^{3} x^{6}+57915 a \,c^{2} d e \,f^{2} x^{6}+57915 a c \,d^{2} e^{2} f \,x^{6}+6435 a \,d^{3} e^{3} x^{6}+19305 b \,c^{3} e \,f^{2} x^{6}+57915 b \,c^{2} d \,e^{2} f \,x^{6}+19305 b c \,d^{2} e^{3} x^{6}+27027 a \,c^{3} e \,f^{2} x^{4}+81081 a \,c^{2} d \,e^{2} f \,x^{4}+27027 a c \,d^{2} e^{3} x^{4}+27027 b \,c^{3} e^{2} f \,x^{4}+27027 b \,c^{2} d \,e^{3} x^{4}+45045 a \,c^{3} e^{2} f \,x^{2}+45045 a \,c^{2} d \,e^{3} x^{2}+15015 b \,c^{3} e^{3} x^{2}+45045 a \,c^{3} e^{3}\right )}{45045} \] Input:

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

Output:

(x*(45045*a*c**3*e**3 + 45045*a*c**3*e**2*f*x**2 + 27027*a*c**3*e*f**2*x** 
4 + 6435*a*c**3*f**3*x**6 + 45045*a*c**2*d*e**3*x**2 + 81081*a*c**2*d*e**2 
*f*x**4 + 57915*a*c**2*d*e*f**2*x**6 + 15015*a*c**2*d*f**3*x**8 + 27027*a* 
c*d**2*e**3*x**4 + 57915*a*c*d**2*e**2*f*x**6 + 45045*a*c*d**2*e*f**2*x**8 
 + 12285*a*c*d**2*f**3*x**10 + 6435*a*d**3*e**3*x**6 + 15015*a*d**3*e**2*f 
*x**8 + 12285*a*d**3*e*f**2*x**10 + 3465*a*d**3*f**3*x**12 + 15015*b*c**3* 
e**3*x**2 + 27027*b*c**3*e**2*f*x**4 + 19305*b*c**3*e*f**2*x**6 + 5005*b*c 
**3*f**3*x**8 + 27027*b*c**2*d*e**3*x**4 + 57915*b*c**2*d*e**2*f*x**6 + 45 
045*b*c**2*d*e*f**2*x**8 + 12285*b*c**2*d*f**3*x**10 + 19305*b*c*d**2*e**3 
*x**6 + 45045*b*c*d**2*e**2*f*x**8 + 36855*b*c*d**2*e*f**2*x**10 + 10395*b 
*c*d**2*f**3*x**12 + 5005*b*d**3*e**3*x**8 + 12285*b*d**3*e**2*f*x**10 + 1 
0395*b*d**3*e*f**2*x**12 + 3003*b*d**3*f**3*x**14))/45045