\(\int (A+B x+C x^2) (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3)^2 \, dx\) [110]

Optimal result

Integrand size = 57, antiderivative size = 664 \[ \int \left (A+B x+C x^2\right ) \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=\frac {\left (A b^2-a (b B-a C)\right ) (b c-a d)^2 (b e-a f)^2 (a+b x)^3}{3 b^7}-\frac {(b c-a d) (b e-a f) \left (6 a^3 C d f+a b^2 (2 c C e+3 B d e+3 B c f+4 A d f)-b^3 (B c e+2 A (d e+c f))-a^2 b (5 B d f+4 C (d e+c f))\right ) (a+b x)^4}{4 b^7}+\frac {\left (15 a^4 C d^2 f^2-10 a^3 b d f (B d f+2 C (d e+c f))+b^4 \left (A d^2 e^2+2 c d e (B e+2 A f)+c^2 \left (C e^2+2 B e f+A f^2\right )\right )-3 a b^3 \left (d^2 e (B e+2 A f)+c^2 f (2 C e+B f)+2 c d \left (C e^2+2 B e f+A f^2\right )\right )+6 a^2 b^2 \left (d f (2 B d e+2 B c f+A d f)+C \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right )\right ) (a+b x)^5}{5 b^7}-\frac {\left (20 a^3 C d^2 f^2-10 a^2 b d f (B d f+2 C (d e+c f))-b^3 \left (d^2 e (B e+2 A f)+c^2 f (2 C e+B f)+2 c d \left (C e^2+2 B e f+A f^2\right )\right )+4 a b^2 \left (d f (2 B d e+2 B c f+A d f)+C \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right )\right ) (a+b x)^6}{6 b^7}+\frac {\left (15 a^2 C d^2 f^2-5 a b d f (B d f+2 C (d e+c f))+b^2 \left (d f (2 B d e+2 B c f+A d f)+C \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right )\right ) (a+b x)^7}{7 b^7}-\frac {d f (6 a C d f-b (B d f+2 C (d e+c f))) (a+b x)^8}{8 b^7}+\frac {C d^2 f^2 (a+b x)^9}{9 b^7} \] Output:

1/3*(A*b^2-a*(B*b-C*a))*(-a*d+b*c)^2*(-a*f+b*e)^2*(b*x+a)^3/b^7-1/4*(-a*d+ 
b*c)*(-a*f+b*e)*(6*a^3*C*d*f+a*b^2*(4*A*d*f+3*B*c*f+3*B*d*e+2*C*c*e)-b^3*( 
B*c*e+2*A*(c*f+d*e))-a^2*b*(5*B*d*f+4*C*(c*f+d*e)))*(b*x+a)^4/b^7+1/5*(15* 
a^4*C*d^2*f^2-10*a^3*b*d*f*(B*d*f+2*C*(c*f+d*e))+b^4*(A*d^2*e^2+2*c*d*e*(2 
*A*f+B*e)+c^2*(A*f^2+2*B*e*f+C*e^2))-3*a*b^3*(d^2*e*(2*A*f+B*e)+c^2*f*(B*f 
+2*C*e)+2*c*d*(A*f^2+2*B*e*f+C*e^2))+6*a^2*b^2*(d*f*(A*d*f+2*B*c*f+2*B*d*e 
)+C*(c^2*f^2+4*c*d*e*f+d^2*e^2)))*(b*x+a)^5/b^7-1/6*(20*a^3*C*d^2*f^2-10*a 
^2*b*d*f*(B*d*f+2*C*(c*f+d*e))-b^3*(d^2*e*(2*A*f+B*e)+c^2*f*(B*f+2*C*e)+2* 
c*d*(A*f^2+2*B*e*f+C*e^2))+4*a*b^2*(d*f*(A*d*f+2*B*c*f+2*B*d*e)+C*(c^2*f^2 
+4*c*d*e*f+d^2*e^2)))*(b*x+a)^6/b^7+1/7*(15*a^2*C*d^2*f^2-5*a*b*d*f*(B*d*f 
+2*C*(c*f+d*e))+b^2*(d*f*(A*d*f+2*B*c*f+2*B*d*e)+C*(c^2*f^2+4*c*d*e*f+d^2* 
e^2)))*(b*x+a)^7/b^7-1/8*d*f*(6*a*C*d*f-b*(B*d*f+2*C*(c*f+d*e)))*(b*x+a)^8 
/b^7+1/9*C*d^2*f^2*(b*x+a)^9/b^7
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.01 \[ \int \left (A+B x+C x^2\right ) \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=a^2 A c^2 e^2 x+\frac {1}{2} a c e (a B c e+2 A (b c e+a d e+a c f)) x^2+\frac {1}{3} \left (a c e (2 b B c e+a (c C e+2 B d e+2 B c f))+A \left (b^2 c^2 e^2+4 a b c e (d e+c f)+a^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right )\right ) x^3+\frac {1}{4} \left (b^2 c e (B c e+2 A (d e+c f))+2 a b \left (A d^2 e^2+2 c d e (B e+2 A f)+c^2 \left (C e^2+2 B e f+A f^2\right )\right )+a^2 \left (d^2 e (B e+2 A f)+c^2 f (2 C e+B f)+2 c d \left (C e^2+2 B e f+A f^2\right )\right )\right ) x^4+\frac {1}{5} \left (b^2 \left (A d^2 e^2+2 c d e (B e+2 A f)+c^2 \left (C e^2+2 B e f+A f^2\right )\right )+2 a b \left (d^2 e (B e+2 A f)+c^2 f (2 C e+B f)+2 c d \left (C e^2+2 B e f+A f^2\right )\right )+a^2 \left (d f (2 B d e+2 B c f+A d f)+C \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right )\right ) x^5+\frac {1}{6} \left (a^2 d f (B d f+2 C (d e+c f))+b^2 \left (d^2 e (B e+2 A f)+c^2 f (2 C e+B f)+2 c d \left (C e^2+2 B e f+A f^2\right )\right )+2 a b \left (d f (2 B d e+2 B c f+A d f)+C \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right )\right ) x^6+\frac {1}{7} \left (a^2 C d^2 f^2+2 a b d f (B d f+2 C (d e+c f))+b^2 \left (d f (2 B d e+2 B c f+A d f)+C \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right )\right ) x^7+\frac {1}{8} b d f (b B d f+2 a C d f+2 b C (d e+c f)) x^8+\frac {1}{9} b^2 C d^2 f^2 x^9 \] Input:

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

Output:

a^2*A*c^2*e^2*x + (a*c*e*(a*B*c*e + 2*A*(b*c*e + a*d*e + a*c*f))*x^2)/2 + 
((a*c*e*(2*b*B*c*e + a*(c*C*e + 2*B*d*e + 2*B*c*f)) + A*(b^2*c^2*e^2 + 4*a 
*b*c*e*(d*e + c*f) + a^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)))*x^3)/3 + ((b^2* 
c*e*(B*c*e + 2*A*(d*e + c*f)) + 2*a*b*(A*d^2*e^2 + 2*c*d*e*(B*e + 2*A*f) + 
 c^2*(C*e^2 + 2*B*e*f + A*f^2)) + a^2*(d^2*e*(B*e + 2*A*f) + c^2*f*(2*C*e 
+ B*f) + 2*c*d*(C*e^2 + 2*B*e*f + A*f^2)))*x^4)/4 + ((b^2*(A*d^2*e^2 + 2*c 
*d*e*(B*e + 2*A*f) + c^2*(C*e^2 + 2*B*e*f + A*f^2)) + 2*a*b*(d^2*e*(B*e + 
2*A*f) + c^2*f*(2*C*e + B*f) + 2*c*d*(C*e^2 + 2*B*e*f + A*f^2)) + a^2*(d*f 
*(2*B*d*e + 2*B*c*f + A*d*f) + C*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)))*x^5)/5 
+ ((a^2*d*f*(B*d*f + 2*C*(d*e + c*f)) + b^2*(d^2*e*(B*e + 2*A*f) + c^2*f*( 
2*C*e + B*f) + 2*c*d*(C*e^2 + 2*B*e*f + A*f^2)) + 2*a*b*(d*f*(2*B*d*e + 2* 
B*c*f + A*d*f) + C*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)))*x^6)/6 + ((a^2*C*d^2* 
f^2 + 2*a*b*d*f*(B*d*f + 2*C*(d*e + c*f)) + b^2*(d*f*(2*B*d*e + 2*B*c*f + 
A*d*f) + C*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)))*x^7)/7 + (b*d*f*(b*B*d*f + 2* 
a*C*d*f + 2*b*C*(d*e + c*f))*x^8)/8 + (b^2*C*d^2*f^2*x^9)/9
 

Rubi [A] (verified)

Time = 2.51 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {2188, 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.

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

\(\Big \downarrow \) 2188

\(\displaystyle \int \left (x^6 \left (a^2 C d^2 f^2+2 a b d f (B d f+2 C (c f+d e))+b^2 \left (d f (A d f+2 B c f+2 B d e)+C \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )\right )+x^5 \left (a^2 d f (B d f+2 C (c f+d e))+2 a b \left (d f (A d f+2 B c f+2 B d e)+C \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+b^2 \left (2 c d \left (A f^2+2 B e f+C e^2\right )+d^2 e (2 A f+B e)+c^2 f (B f+2 C e)\right )\right )+x^4 \left (a^2 \left (d f (A d f+2 B c f+2 B d e)+C \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+2 a b \left (2 c d \left (A f^2+2 B e f+C e^2\right )+d^2 e (2 A f+B e)+c^2 f (B f+2 C e)\right )+b^2 \left (c^2 \left (A f^2+2 B e f+C e^2\right )+2 c d e (2 A f+B e)+A d^2 e^2\right )\right )+x^3 \left (a^2 \left (2 c d \left (A f^2+2 B e f+C e^2\right )+d^2 e (2 A f+B e)+c^2 f (B f+2 C e)\right )+2 a b \left (c^2 \left (A f^2+2 B e f+C e^2\right )+2 c d e (2 A f+B e)+A d^2 e^2\right )+b^2 c e (2 A (c f+d e)+B c e)\right )+x^2 \left (A \left (a^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )+4 a b c e (c f+d e)+b^2 c^2 e^2\right )+a c e (a (2 B c f+2 B d e+c C e)+2 b B c e)\right )+a^2 A c^2 e^2+a c e x (2 A (a c f+a d e+b c e)+a B c e)+b d f x^7 (2 a C d f+b B d f+2 b C (c f+d e))+b^2 C d^2 f^2 x^8\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{7} x^7 \left (a^2 C d^2 f^2+2 a b d f (B d f+2 C (c f+d e))+b^2 \left (d f (A d f+2 B c f+2 B d e)+C \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )\right )+\frac {1}{6} x^6 \left (a^2 d f (B d f+2 C (c f+d e))+2 a b \left (d f (A d f+2 B c f+2 B d e)+C \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+b^2 \left (2 c d \left (A f^2+2 B e f+C e^2\right )+d^2 e (2 A f+B e)+c^2 f (B f+2 C e)\right )\right )+\frac {1}{5} x^5 \left (a^2 \left (d f (A d f+2 B c f+2 B d e)+C \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+2 a b \left (2 c d \left (A f^2+2 B e f+C e^2\right )+d^2 e (2 A f+B e)+c^2 f (B f+2 C e)\right )+b^2 \left (c^2 \left (A f^2+2 B e f+C e^2\right )+2 c d e (2 A f+B e)+A d^2 e^2\right )\right )+\frac {1}{4} x^4 \left (a^2 \left (2 c d \left (A f^2+2 B e f+C e^2\right )+d^2 e (2 A f+B e)+c^2 f (B f+2 C e)\right )+2 a b \left (c^2 \left (A f^2+2 B e f+C e^2\right )+2 c d e (2 A f+B e)+A d^2 e^2\right )+b^2 c e (2 A (c f+d e)+B c e)\right )+\frac {1}{3} x^3 \left (A \left (a^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )+4 a b c e (c f+d e)+b^2 c^2 e^2\right )+a c e (a (2 B c f+2 B d e+c C e)+2 b B c e)\right )+a^2 A c^2 e^2 x+\frac {1}{2} a c e x^2 (2 A (a c f+a d e+b c e)+a B c e)+\frac {1}{8} b d f x^8 (2 a C d f+b B d f+2 b C (c f+d e))+\frac {1}{9} b^2 C d^2 f^2 x^9\)

Input:

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

Output:

a^2*A*c^2*e^2*x + (a*c*e*(a*B*c*e + 2*A*(b*c*e + a*d*e + a*c*f))*x^2)/2 + 
((a*c*e*(2*b*B*c*e + a*(c*C*e + 2*B*d*e + 2*B*c*f)) + A*(b^2*c^2*e^2 + 4*a 
*b*c*e*(d*e + c*f) + a^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)))*x^3)/3 + ((b^2* 
c*e*(B*c*e + 2*A*(d*e + c*f)) + 2*a*b*(A*d^2*e^2 + 2*c*d*e*(B*e + 2*A*f) + 
 c^2*(C*e^2 + 2*B*e*f + A*f^2)) + a^2*(d^2*e*(B*e + 2*A*f) + c^2*f*(2*C*e 
+ B*f) + 2*c*d*(C*e^2 + 2*B*e*f + A*f^2)))*x^4)/4 + ((b^2*(A*d^2*e^2 + 2*c 
*d*e*(B*e + 2*A*f) + c^2*(C*e^2 + 2*B*e*f + A*f^2)) + 2*a*b*(d^2*e*(B*e + 
2*A*f) + c^2*f*(2*C*e + B*f) + 2*c*d*(C*e^2 + 2*B*e*f + A*f^2)) + a^2*(d*f 
*(2*B*d*e + 2*B*c*f + A*d*f) + C*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)))*x^5)/5 
+ ((a^2*d*f*(B*d*f + 2*C*(d*e + c*f)) + b^2*(d^2*e*(B*e + 2*A*f) + c^2*f*( 
2*C*e + B*f) + 2*c*d*(C*e^2 + 2*B*e*f + A*f^2)) + 2*a*b*(d*f*(2*B*d*e + 2* 
B*c*f + A*d*f) + C*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)))*x^6)/6 + ((a^2*C*d^2* 
f^2 + 2*a*b*d*f*(B*d*f + 2*C*(d*e + c*f)) + b^2*(d*f*(2*B*d*e + 2*B*c*f + 
A*d*f) + C*(d^2*e^2 + 4*c*d*e*f + c^2*f^2)))*x^7)/7 + (b*d*f*(b*B*d*f + 2* 
a*C*d*f + 2*b*C*(d*e + c*f))*x^8)/8 + (b^2*C*d^2*f^2*x^9)/9
 

Defintions of rubi rules used

rule 2009

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

rule 2188

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq 
, x] && IGtQ[p, -2]
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 562, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {C \,f^{2} d^{2} b^{2} x^{9}}{9}+\frac {\left (B \,b^{2} d^{2} f^{2}+2 C \left (a d f +b c f +b d e \right ) b d f \right ) x^{8}}{8}+\frac {\left (A \,b^{2} d^{2} f^{2}+2 B \left (a d f +b c f +b d e \right ) b d f +C \left (2 \left (a c f +a d e +b c e \right ) b d f +\left (a d f +b c f +b d e \right )^{2}\right )\right ) x^{7}}{7}+\frac {\left (2 A \left (a d f +b c f +b d e \right ) b d f +B \left (2 \left (a c f +a d e +b c e \right ) b d f +\left (a d f +b c f +b d e \right )^{2}\right )+C \left (2 a c e b d f +2 \left (a c f +a d e +b c e \right ) \left (a d f +b c f +b d e \right )\right )\right ) x^{6}}{6}+\frac {\left (A \left (2 \left (a c f +a d e +b c e \right ) b d f +\left (a d f +b c f +b d e \right )^{2}\right )+B \left (2 a c e b d f +2 \left (a c f +a d e +b c e \right ) \left (a d f +b c f +b d e \right )\right )+C \left (2 a c e \left (a d f +b c f +b d e \right )+\left (a c f +a d e +b c e \right )^{2}\right )\right ) x^{5}}{5}+\frac {\left (A \left (2 a c e b d f +2 \left (a c f +a d e +b c e \right ) \left (a d f +b c f +b d e \right )\right )+B \left (2 a c e \left (a d f +b c f +b d e \right )+\left (a c f +a d e +b c e \right )^{2}\right )+2 C a c e \left (a c f +a d e +b c e \right )\right ) x^{4}}{4}+\frac {\left (A \left (2 a c e \left (a d f +b c f +b d e \right )+\left (a c f +a d e +b c e \right )^{2}\right )+2 B a c e \left (a c f +a d e +b c e \right )+C \,a^{2} c^{2} e^{2}\right ) x^{3}}{3}+\frac {\left (2 A a c e \left (a c f +a d e +b c e \right )+B \,a^{2} c^{2} e^{2}\right ) x^{2}}{2}+A \,c^{2} e^{2} a^{2} x\)	\(562\)
norman	\(\frac {C \,f^{2} d^{2} b^{2} x^{9}}{9}+\left (\frac {1}{8} B \,b^{2} d^{2} f^{2}+\frac {1}{4} C \,f^{2} d^{2} a b +\frac {1}{4} C \,b^{2} c d \,f^{2}+\frac {1}{4} C \,b^{2} d^{2} e f \right ) x^{8}+\left (\frac {1}{7} A \,b^{2} d^{2} f^{2}+\frac {2}{7} B a b \,d^{2} f^{2}+\frac {2}{7} B \,b^{2} c d \,f^{2}+\frac {2}{7} B \,b^{2} d^{2} e f +\frac {1}{7} a^{2} C \,d^{2} f^{2}+\frac {4}{7} C a b c d \,f^{2}+\frac {4}{7} C a b \,d^{2} e f +\frac {1}{7} C \,b^{2} c^{2} f^{2}+\frac {4}{7} C \,b^{2} c d e f +\frac {1}{7} C \,b^{2} d^{2} e^{2}\right ) x^{7}+\left (\frac {1}{3} A a b \,d^{2} f^{2}+\frac {1}{3} A \,b^{2} c d \,f^{2}+\frac {1}{3} A \,b^{2} d^{2} e f +\frac {1}{6} B \,a^{2} d^{2} f^{2}+\frac {2}{3} B a b c d \,f^{2}+\frac {2}{3} B a b \,d^{2} e f +\frac {1}{6} B \,b^{2} c^{2} f^{2}+\frac {2}{3} B \,b^{2} c d e f +\frac {1}{6} B \,b^{2} d^{2} e^{2}+\frac {1}{3} C \,a^{2} c d \,f^{2}+\frac {1}{3} C \,a^{2} d^{2} e f +\frac {1}{3} C a b \,c^{2} f^{2}+\frac {4}{3} C a b c d e f +\frac {1}{3} C a b \,d^{2} e^{2}+\frac {1}{3} C \,b^{2} c^{2} e f +\frac {1}{3} C \,b^{2} c d \,e^{2}\right ) x^{6}+\left (\frac {1}{5} A \,a^{2} d^{2} f^{2}+\frac {4}{5} A a b c d \,f^{2}+\frac {4}{5} A a b \,d^{2} e f +\frac {1}{5} A \,b^{2} c^{2} f^{2}+\frac {4}{5} A \,b^{2} c d e f +\frac {1}{5} A \,b^{2} d^{2} e^{2}+\frac {2}{5} B \,a^{2} c d \,f^{2}+\frac {2}{5} B \,a^{2} d^{2} e f +\frac {2}{5} B a b \,c^{2} f^{2}+\frac {8}{5} B a b c d e f +\frac {2}{5} B a b \,d^{2} e^{2}+\frac {2}{5} B \,b^{2} c^{2} e f +\frac {2}{5} B \,b^{2} c d \,e^{2}+\frac {1}{5} C \,a^{2} c^{2} f^{2}+\frac {4}{5} C \,a^{2} c d e f +\frac {1}{5} C \,a^{2} d^{2} e^{2}+\frac {4}{5} C a b \,c^{2} e f +\frac {4}{5} C a b c d \,e^{2}+\frac {1}{5} C \,b^{2} c^{2} e^{2}\right ) x^{5}+\left (\frac {1}{2} A \,a^{2} c d \,f^{2}+\frac {1}{2} A \,a^{2} d^{2} e f +\frac {1}{2} A a b \,c^{2} f^{2}+2 A a b c d e f +\frac {1}{2} A a b \,d^{2} e^{2}+\frac {1}{2} A \,b^{2} c^{2} e f +\frac {1}{2} A \,b^{2} c d \,e^{2}+\frac {1}{4} B \,a^{2} c^{2} f^{2}+B \,a^{2} c d e f +\frac {1}{4} B \,a^{2} d^{2} e^{2}+B a b \,c^{2} e f +B a b c d \,e^{2}+\frac {1}{4} B \,b^{2} c^{2} e^{2}+\frac {1}{2} C \,a^{2} c^{2} e f +\frac {1}{2} C \,a^{2} c d \,e^{2}+\frac {1}{2} C a b \,c^{2} e^{2}\right ) x^{4}+\left (\frac {1}{3} A \,a^{2} c^{2} f^{2}+\frac {4}{3} A \,a^{2} c d e f +\frac {1}{3} A \,a^{2} d^{2} e^{2}+\frac {4}{3} A a b \,c^{2} e f +\frac {4}{3} A a b c d \,e^{2}+\frac {1}{3} A \,c^{2} e^{2} b^{2}+\frac {2}{3} B \,a^{2} c^{2} e f +\frac {2}{3} B \,a^{2} c d \,e^{2}+\frac {2}{3} B a b \,c^{2} e^{2}+\frac {1}{3} C \,a^{2} c^{2} e^{2}\right ) x^{3}+\left (A \,a^{2} c^{2} e f +A \,a^{2} c d \,e^{2}+A \,c^{2} e^{2} a b +\frac {1}{2} B \,a^{2} c^{2} e^{2}\right ) x^{2}+A \,c^{2} e^{2} a^{2} x\)	\(925\)
risch	\(\text {Expression too large to display}\)	\(1127\)
parallelrisch	\(\text {Expression too large to display}\)	\(1127\)
gosper	\(\text {Expression too large to display}\)	\(1128\)
orering	\(\text {Expression too large to display}\)	\(1195\)

Input:

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

Output:

1/9*C*f^2*d^2*b^2*x^9+1/8*(B*b^2*d^2*f^2+2*C*(a*d*f+b*c*f+b*d*e)*b*d*f)*x^ 
8+1/7*(A*b^2*d^2*f^2+2*B*(a*d*f+b*c*f+b*d*e)*b*d*f+C*(2*(a*c*f+a*d*e+b*c*e 
)*b*d*f+(a*d*f+b*c*f+b*d*e)^2))*x^7+1/6*(2*A*(a*d*f+b*c*f+b*d*e)*b*d*f+B*( 
2*(a*c*f+a*d*e+b*c*e)*b*d*f+(a*d*f+b*c*f+b*d*e)^2)+C*(2*a*c*e*b*d*f+2*(a*c 
*f+a*d*e+b*c*e)*(a*d*f+b*c*f+b*d*e)))*x^6+1/5*(A*(2*(a*c*f+a*d*e+b*c*e)*b* 
d*f+(a*d*f+b*c*f+b*d*e)^2)+B*(2*a*c*e*b*d*f+2*(a*c*f+a*d*e+b*c*e)*(a*d*f+b 
*c*f+b*d*e))+C*(2*a*c*e*(a*d*f+b*c*f+b*d*e)+(a*c*f+a*d*e+b*c*e)^2))*x^5+1/ 
4*(A*(2*a*c*e*b*d*f+2*(a*c*f+a*d*e+b*c*e)*(a*d*f+b*c*f+b*d*e))+B*(2*a*c*e* 
(a*d*f+b*c*f+b*d*e)+(a*c*f+a*d*e+b*c*e)^2)+2*C*a*c*e*(a*c*f+a*d*e+b*c*e))* 
x^4+1/3*(A*(2*a*c*e*(a*d*f+b*c*f+b*d*e)+(a*c*f+a*d*e+b*c*e)^2)+2*B*a*c*e*( 
a*c*f+a*d*e+b*c*e)+C*a^2*c^2*e^2)*x^3+1/2*(2*A*a*c*e*(a*c*f+a*d*e+b*c*e)+B 
*a^2*c^2*e^2)*x^2+A*c^2*e^2*a^2*x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.11 \[ \int \left (A+B x+C x^2\right ) \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

1/9*C*b^2*d^2*f^2*x^9 + 1/8*(2*C*b^2*d^2*e*f + (2*C*b^2*c*d + (2*C*a*b + B 
*b^2)*d^2)*f^2)*x^8 + A*a^2*c^2*e^2*x + 1/7*(C*b^2*d^2*e^2 + 2*(2*C*b^2*c* 
d + (2*C*a*b + B*b^2)*d^2)*e*f + (C*b^2*c^2 + 2*(2*C*a*b + B*b^2)*c*d + (C 
*a^2 + 2*B*a*b + A*b^2)*d^2)*f^2)*x^7 + 1/6*((2*C*b^2*c*d + (2*C*a*b + B*b 
^2)*d^2)*e^2 + 2*(C*b^2*c^2 + 2*(2*C*a*b + B*b^2)*c*d + (C*a^2 + 2*B*a*b + 
 A*b^2)*d^2)*e*f + ((2*C*a*b + B*b^2)*c^2 + 2*(C*a^2 + 2*B*a*b + A*b^2)*c* 
d + (B*a^2 + 2*A*a*b)*d^2)*f^2)*x^6 + 1/5*((C*b^2*c^2 + 2*(2*C*a*b + B*b^2 
)*c*d + (C*a^2 + 2*B*a*b + A*b^2)*d^2)*e^2 + 2*((2*C*a*b + B*b^2)*c^2 + 2* 
(C*a^2 + 2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*e*f + (A*a^2*d^2 + 
(C*a^2 + 2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*f^2)*x^5 + 1/4*(( 
(2*C*a*b + B*b^2)*c^2 + 2*(C*a^2 + 2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b 
)*d^2)*e^2 + 2*(A*a^2*d^2 + (C*a^2 + 2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A 
*a*b)*c*d)*e*f + (2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*f^2)*x^4 + 1/3*(A*a 
^2*c^2*f^2 + (A*a^2*d^2 + (C*a^2 + 2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a 
*b)*c*d)*e^2 + 2*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*e*f)*x^3 + 1/2*(2*A 
*a^2*c^2*e*f + (2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*e^2)*x^2
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 1166, normalized size of antiderivative = 1.76 \[ \int \left (A+B x+C x^2\right ) \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

A*a**2*c**2*e**2*x + C*b**2*d**2*f**2*x**9/9 + x**8*(B*b**2*d**2*f**2/8 + 
C*a*b*d**2*f**2/4 + C*b**2*c*d*f**2/4 + C*b**2*d**2*e*f/4) + x**7*(A*b**2* 
d**2*f**2/7 + 2*B*a*b*d**2*f**2/7 + 2*B*b**2*c*d*f**2/7 + 2*B*b**2*d**2*e* 
f/7 + C*a**2*d**2*f**2/7 + 4*C*a*b*c*d*f**2/7 + 4*C*a*b*d**2*e*f/7 + C*b** 
2*c**2*f**2/7 + 4*C*b**2*c*d*e*f/7 + C*b**2*d**2*e**2/7) + x**6*(A*a*b*d** 
2*f**2/3 + A*b**2*c*d*f**2/3 + A*b**2*d**2*e*f/3 + B*a**2*d**2*f**2/6 + 2* 
B*a*b*c*d*f**2/3 + 2*B*a*b*d**2*e*f/3 + B*b**2*c**2*f**2/6 + 2*B*b**2*c*d* 
e*f/3 + B*b**2*d**2*e**2/6 + C*a**2*c*d*f**2/3 + C*a**2*d**2*e*f/3 + C*a*b 
*c**2*f**2/3 + 4*C*a*b*c*d*e*f/3 + C*a*b*d**2*e**2/3 + C*b**2*c**2*e*f/3 + 
 C*b**2*c*d*e**2/3) + x**5*(A*a**2*d**2*f**2/5 + 4*A*a*b*c*d*f**2/5 + 4*A* 
a*b*d**2*e*f/5 + A*b**2*c**2*f**2/5 + 4*A*b**2*c*d*e*f/5 + A*b**2*d**2*e** 
2/5 + 2*B*a**2*c*d*f**2/5 + 2*B*a**2*d**2*e*f/5 + 2*B*a*b*c**2*f**2/5 + 8* 
B*a*b*c*d*e*f/5 + 2*B*a*b*d**2*e**2/5 + 2*B*b**2*c**2*e*f/5 + 2*B*b**2*c*d 
*e**2/5 + C*a**2*c**2*f**2/5 + 4*C*a**2*c*d*e*f/5 + C*a**2*d**2*e**2/5 + 4 
*C*a*b*c**2*e*f/5 + 4*C*a*b*c*d*e**2/5 + C*b**2*c**2*e**2/5) + x**4*(A*a** 
2*c*d*f**2/2 + A*a**2*d**2*e*f/2 + A*a*b*c**2*f**2/2 + 2*A*a*b*c*d*e*f + A 
*a*b*d**2*e**2/2 + A*b**2*c**2*e*f/2 + A*b**2*c*d*e**2/2 + B*a**2*c**2*f** 
2/4 + B*a**2*c*d*e*f + B*a**2*d**2*e**2/4 + B*a*b*c**2*e*f + B*a*b*c*d*e** 
2 + B*b**2*c**2*e**2/4 + C*a**2*c**2*e*f/2 + C*a**2*c*d*e**2/2 + C*a*b*c** 
2*e**2/2) + x**3*(A*a**2*c**2*f**2/3 + 4*A*a**2*c*d*e*f/3 + A*a**2*d**2...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.11 \[ \int \left (A+B x+C x^2\right ) \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

1/9*C*b^2*d^2*f^2*x^9 + 1/8*(2*C*b^2*d^2*e*f + (2*C*b^2*c*d + (2*C*a*b + B 
*b^2)*d^2)*f^2)*x^8 + A*a^2*c^2*e^2*x + 1/7*(C*b^2*d^2*e^2 + 2*(2*C*b^2*c* 
d + (2*C*a*b + B*b^2)*d^2)*e*f + (C*b^2*c^2 + 2*(2*C*a*b + B*b^2)*c*d + (C 
*a^2 + 2*B*a*b + A*b^2)*d^2)*f^2)*x^7 + 1/6*((2*C*b^2*c*d + (2*C*a*b + B*b 
^2)*d^2)*e^2 + 2*(C*b^2*c^2 + 2*(2*C*a*b + B*b^2)*c*d + (C*a^2 + 2*B*a*b + 
 A*b^2)*d^2)*e*f + ((2*C*a*b + B*b^2)*c^2 + 2*(C*a^2 + 2*B*a*b + A*b^2)*c* 
d + (B*a^2 + 2*A*a*b)*d^2)*f^2)*x^6 + 1/5*((C*b^2*c^2 + 2*(2*C*a*b + B*b^2 
)*c*d + (C*a^2 + 2*B*a*b + A*b^2)*d^2)*e^2 + 2*((2*C*a*b + B*b^2)*c^2 + 2* 
(C*a^2 + 2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*e*f + (A*a^2*d^2 + 
(C*a^2 + 2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*f^2)*x^5 + 1/4*(( 
(2*C*a*b + B*b^2)*c^2 + 2*(C*a^2 + 2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b 
)*d^2)*e^2 + 2*(A*a^2*d^2 + (C*a^2 + 2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A 
*a*b)*c*d)*e*f + (2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*f^2)*x^4 + 1/3*(A*a 
^2*c^2*f^2 + (A*a^2*d^2 + (C*a^2 + 2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a 
*b)*c*d)*e^2 + 2*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*e*f)*x^3 + 1/2*(2*A 
*a^2*c^2*e*f + (2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*e^2)*x^2
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 1126, normalized size of antiderivative = 1.70 \[ \int \left (A+B x+C x^2\right ) \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/9*C*b^2*d^2*f^2*x^9 + 1/4*C*b^2*d^2*e*f*x^8 + 1/4*C*b^2*c*d*f^2*x^8 + 1/ 
4*C*a*b*d^2*f^2*x^8 + 1/8*B*b^2*d^2*f^2*x^8 + 1/7*C*b^2*d^2*e^2*x^7 + 4/7* 
C*b^2*c*d*e*f*x^7 + 4/7*C*a*b*d^2*e*f*x^7 + 2/7*B*b^2*d^2*e*f*x^7 + 1/7*C* 
b^2*c^2*f^2*x^7 + 4/7*C*a*b*c*d*f^2*x^7 + 2/7*B*b^2*c*d*f^2*x^7 + 1/7*C*a^ 
2*d^2*f^2*x^7 + 2/7*B*a*b*d^2*f^2*x^7 + 1/7*A*b^2*d^2*f^2*x^7 + 1/3*C*b^2* 
c*d*e^2*x^6 + 1/3*C*a*b*d^2*e^2*x^6 + 1/6*B*b^2*d^2*e^2*x^6 + 1/3*C*b^2*c^ 
2*e*f*x^6 + 4/3*C*a*b*c*d*e*f*x^6 + 2/3*B*b^2*c*d*e*f*x^6 + 1/3*C*a^2*d^2* 
e*f*x^6 + 2/3*B*a*b*d^2*e*f*x^6 + 1/3*A*b^2*d^2*e*f*x^6 + 1/3*C*a*b*c^2*f^ 
2*x^6 + 1/6*B*b^2*c^2*f^2*x^6 + 1/3*C*a^2*c*d*f^2*x^6 + 2/3*B*a*b*c*d*f^2* 
x^6 + 1/3*A*b^2*c*d*f^2*x^6 + 1/6*B*a^2*d^2*f^2*x^6 + 1/3*A*a*b*d^2*f^2*x^ 
6 + 1/5*C*b^2*c^2*e^2*x^5 + 4/5*C*a*b*c*d*e^2*x^5 + 2/5*B*b^2*c*d*e^2*x^5 
+ 1/5*C*a^2*d^2*e^2*x^5 + 2/5*B*a*b*d^2*e^2*x^5 + 1/5*A*b^2*d^2*e^2*x^5 + 
4/5*C*a*b*c^2*e*f*x^5 + 2/5*B*b^2*c^2*e*f*x^5 + 4/5*C*a^2*c*d*e*f*x^5 + 8/ 
5*B*a*b*c*d*e*f*x^5 + 4/5*A*b^2*c*d*e*f*x^5 + 2/5*B*a^2*d^2*e*f*x^5 + 4/5* 
A*a*b*d^2*e*f*x^5 + 1/5*C*a^2*c^2*f^2*x^5 + 2/5*B*a*b*c^2*f^2*x^5 + 1/5*A* 
b^2*c^2*f^2*x^5 + 2/5*B*a^2*c*d*f^2*x^5 + 4/5*A*a*b*c*d*f^2*x^5 + 1/5*A*a^ 
2*d^2*f^2*x^5 + 1/2*C*a*b*c^2*e^2*x^4 + 1/4*B*b^2*c^2*e^2*x^4 + 1/2*C*a^2* 
c*d*e^2*x^4 + B*a*b*c*d*e^2*x^4 + 1/2*A*b^2*c*d*e^2*x^4 + 1/4*B*a^2*d^2*e^ 
2*x^4 + 1/2*A*a*b*d^2*e^2*x^4 + 1/2*C*a^2*c^2*e*f*x^4 + B*a*b*c^2*e*f*x^4 
+ 1/2*A*b^2*c^2*e*f*x^4 + B*a^2*c*d*e*f*x^4 + 2*A*a*b*c*d*e*f*x^4 + 1/2...
 

Mupad [B] (verification not implemented)

Time = 12.18 (sec) , antiderivative size = 891, normalized size of antiderivative = 1.34 \[ \int \left (A+B x+C x^2\right ) \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

x^5*((A*a^2*d^2*f^2)/5 + (A*b^2*c^2*f^2)/5 + (A*b^2*d^2*e^2)/5 + (C*a^2*c^ 
2*f^2)/5 + (C*a^2*d^2*e^2)/5 + (C*b^2*c^2*e^2)/5 + (2*B*a*b*c^2*f^2)/5 + ( 
2*B*a*b*d^2*e^2)/5 + (2*B*a^2*c*d*f^2)/5 + (2*B*b^2*c*d*e^2)/5 + (2*B*a^2* 
d^2*e*f)/5 + (2*B*b^2*c^2*e*f)/5 + (4*A*a*b*c*d*f^2)/5 + (4*C*a*b*c*d*e^2) 
/5 + (4*A*a*b*d^2*e*f)/5 + (4*C*a*b*c^2*e*f)/5 + (4*A*b^2*c*d*e*f)/5 + (4* 
C*a^2*c*d*e*f)/5 + (8*B*a*b*c*d*e*f)/5) + x^4*((B*a^2*c^2*f^2)/4 + (B*a^2* 
d^2*e^2)/4 + (B*b^2*c^2*e^2)/4 + (A*a*b*c^2*f^2)/2 + (A*a*b*d^2*e^2)/2 + ( 
C*a*b*c^2*e^2)/2 + (A*a^2*c*d*f^2)/2 + (A*b^2*c*d*e^2)/2 + (C*a^2*c*d*e^2) 
/2 + (A*a^2*d^2*e*f)/2 + (A*b^2*c^2*e*f)/2 + (C*a^2*c^2*e*f)/2 + B*a*b*c*d 
*e^2 + B*a*b*c^2*e*f + B*a^2*c*d*e*f + 2*A*a*b*c*d*e*f) + x^6*((B*a^2*d^2* 
f^2)/6 + (B*b^2*c^2*f^2)/6 + (B*b^2*d^2*e^2)/6 + (A*a*b*d^2*f^2)/3 + (C*a* 
b*c^2*f^2)/3 + (C*a*b*d^2*e^2)/3 + (A*b^2*c*d*f^2)/3 + (C*a^2*c*d*f^2)/3 + 
 (C*b^2*c*d*e^2)/3 + (A*b^2*d^2*e*f)/3 + (C*a^2*d^2*e*f)/3 + (C*b^2*c^2*e* 
f)/3 + (2*B*a*b*c*d*f^2)/3 + (2*B*a*b*d^2*e*f)/3 + (2*B*b^2*c*d*e*f)/3 + ( 
4*C*a*b*c*d*e*f)/3) + x^3*((A*a^2*c^2*f^2)/3 + (A*a^2*d^2*e^2)/3 + (A*b^2* 
c^2*e^2)/3 + (C*a^2*c^2*e^2)/3 + (2*B*a*b*c^2*e^2)/3 + (2*B*a^2*c*d*e^2)/3 
 + (2*B*a^2*c^2*e*f)/3 + (4*A*a*b*c*d*e^2)/3 + (4*A*a*b*c^2*e*f)/3 + (4*A* 
a^2*c*d*e*f)/3) + x^7*((A*b^2*d^2*f^2)/7 + (C*a^2*d^2*f^2)/7 + (C*b^2*c^2* 
f^2)/7 + (C*b^2*d^2*e^2)/7 + (2*B*a*b*d^2*f^2)/7 + (2*B*b^2*c*d*f^2)/7 + ( 
2*B*b^2*d^2*e*f)/7 + (4*C*a*b*c*d*f^2)/7 + (4*C*a*b*d^2*e*f)/7 + (4*C*b...
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 871, normalized size of antiderivative = 1.31 \[ \int \left (A+B x+C x^2\right ) \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx=\frac {x \left (280 b^{2} c \,d^{2} f^{2} x^{8}+630 a b c \,d^{2} f^{2} x^{7}+315 b^{3} d^{2} f^{2} x^{7}+630 b^{2} c^{2} d \,f^{2} x^{7}+630 b^{2} c \,d^{2} e f \,x^{7}+360 a^{2} c \,d^{2} f^{2} x^{6}+1080 a \,b^{2} d^{2} f^{2} x^{6}+1440 a b \,c^{2} d \,f^{2} x^{6}+1440 a b c \,d^{2} e f \,x^{6}+720 b^{3} c d \,f^{2} x^{6}+720 b^{3} d^{2} e f \,x^{6}+360 b^{2} c^{3} f^{2} x^{6}+1440 b^{2} c^{2} d e f \,x^{6}+360 b^{2} c \,d^{2} e^{2} x^{6}+1260 a^{2} b \,d^{2} f^{2} x^{5}+840 a^{2} c^{2} d \,f^{2} x^{5}+840 a^{2} c \,d^{2} e f \,x^{5}+2520 a \,b^{2} c d \,f^{2} x^{5}+2520 a \,b^{2} d^{2} e f \,x^{5}+840 a b \,c^{3} f^{2} x^{5}+3360 a b \,c^{2} d e f \,x^{5}+840 a b c \,d^{2} e^{2} x^{5}+420 b^{3} c^{2} f^{2} x^{5}+1680 b^{3} c d e f \,x^{5}+420 b^{3} d^{2} e^{2} x^{5}+840 b^{2} c^{3} e f \,x^{5}+840 b^{2} c^{2} d \,e^{2} x^{5}+504 a^{3} d^{2} f^{2} x^{4}+3024 a^{2} b c d \,f^{2} x^{4}+3024 a^{2} b \,d^{2} e f \,x^{4}+504 a^{2} c^{3} f^{2} x^{4}+2016 a^{2} c^{2} d e f \,x^{4}+504 a^{2} c \,d^{2} e^{2} x^{4}+1512 a \,b^{2} c^{2} f^{2} x^{4}+6048 a \,b^{2} c d e f \,x^{4}+1512 a \,b^{2} d^{2} e^{2} x^{4}+2016 a b \,c^{3} e f \,x^{4}+2016 a b \,c^{2} d \,e^{2} x^{4}+1008 b^{3} c^{2} e f \,x^{4}+1008 b^{3} c d \,e^{2} x^{4}+504 b^{2} c^{3} e^{2} x^{4}+1260 a^{3} c d \,f^{2} x^{3}+1260 a^{3} d^{2} e f \,x^{3}+1890 a^{2} b \,c^{2} f^{2} x^{3}+7560 a^{2} b c d e f \,x^{3}+1890 a^{2} b \,d^{2} e^{2} x^{3}+1260 a^{2} c^{3} e f \,x^{3}+1260 a^{2} c^{2} d \,e^{2} x^{3}+3780 a \,b^{2} c^{2} e f \,x^{3}+3780 a \,b^{2} c d \,e^{2} x^{3}+1260 a b \,c^{3} e^{2} x^{3}+630 b^{3} c^{2} e^{2} x^{3}+840 a^{3} c^{2} f^{2} x^{2}+3360 a^{3} c d e f \,x^{2}+840 a^{3} d^{2} e^{2} x^{2}+5040 a^{2} b \,c^{2} e f \,x^{2}+5040 a^{2} b c d \,e^{2} x^{2}+840 a^{2} c^{3} e^{2} x^{2}+2520 a \,b^{2} c^{2} e^{2} x^{2}+2520 a^{3} c^{2} e f x +2520 a^{3} c d \,e^{2} x +3780 a^{2} b \,c^{2} e^{2} x +2520 a^{3} c^{2} e^{2}\right )}{2520} \] Input:

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

Output:

(x*(2520*a**3*c**2*e**2 + 2520*a**3*c**2*e*f*x + 840*a**3*c**2*f**2*x**2 + 
 2520*a**3*c*d*e**2*x + 3360*a**3*c*d*e*f*x**2 + 1260*a**3*c*d*f**2*x**3 + 
 840*a**3*d**2*e**2*x**2 + 1260*a**3*d**2*e*f*x**3 + 504*a**3*d**2*f**2*x* 
*4 + 3780*a**2*b*c**2*e**2*x + 5040*a**2*b*c**2*e*f*x**2 + 1890*a**2*b*c** 
2*f**2*x**3 + 5040*a**2*b*c*d*e**2*x**2 + 7560*a**2*b*c*d*e*f*x**3 + 3024* 
a**2*b*c*d*f**2*x**4 + 1890*a**2*b*d**2*e**2*x**3 + 3024*a**2*b*d**2*e*f*x 
**4 + 1260*a**2*b*d**2*f**2*x**5 + 840*a**2*c**3*e**2*x**2 + 1260*a**2*c** 
3*e*f*x**3 + 504*a**2*c**3*f**2*x**4 + 1260*a**2*c**2*d*e**2*x**3 + 2016*a 
**2*c**2*d*e*f*x**4 + 840*a**2*c**2*d*f**2*x**5 + 504*a**2*c*d**2*e**2*x** 
4 + 840*a**2*c*d**2*e*f*x**5 + 360*a**2*c*d**2*f**2*x**6 + 2520*a*b**2*c** 
2*e**2*x**2 + 3780*a*b**2*c**2*e*f*x**3 + 1512*a*b**2*c**2*f**2*x**4 + 378 
0*a*b**2*c*d*e**2*x**3 + 6048*a*b**2*c*d*e*f*x**4 + 2520*a*b**2*c*d*f**2*x 
**5 + 1512*a*b**2*d**2*e**2*x**4 + 2520*a*b**2*d**2*e*f*x**5 + 1080*a*b**2 
*d**2*f**2*x**6 + 1260*a*b*c**3*e**2*x**3 + 2016*a*b*c**3*e*f*x**4 + 840*a 
*b*c**3*f**2*x**5 + 2016*a*b*c**2*d*e**2*x**4 + 3360*a*b*c**2*d*e*f*x**5 + 
 1440*a*b*c**2*d*f**2*x**6 + 840*a*b*c*d**2*e**2*x**5 + 1440*a*b*c*d**2*e* 
f*x**6 + 630*a*b*c*d**2*f**2*x**7 + 630*b**3*c**2*e**2*x**3 + 1008*b**3*c* 
*2*e*f*x**4 + 420*b**3*c**2*f**2*x**5 + 1008*b**3*c*d*e**2*x**4 + 1680*b** 
3*c*d*e*f*x**5 + 720*b**3*c*d*f**2*x**6 + 420*b**3*d**2*e**2*x**5 + 720*b* 
*3*d**2*e*f*x**6 + 315*b**3*d**2*f**2*x**7 + 504*b**2*c**3*e**2*x**4 + ...