\(\int (A+B x+C x^2) (a d+(b d+a e) x+(c d+b e) x^2+c e x^3)^2 \, dx\) [125]

Optimal result

Integrand size = 43, antiderivative size = 431 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=a^2 A d^2 x+\frac {1}{2} a d (a B d+2 A (b d+a e)) x^2+\frac {1}{3} \left (a d (2 b B d+a C d+2 a B e)+A \left (b^2 d^2+4 a b d e+a \left (2 c d^2+a e^2\right )\right )\right ) x^3+\frac {1}{4} \left (b^2 d (B d+2 A e)+a \left (2 B c d^2+4 A c d e+2 a C d e+a B e^2\right )+2 b \left (a d (C d+2 B e)+A \left (c d^2+a e^2\right )\right )\right ) x^4+\frac {1}{5} \left (A c \left (c d^2+2 a e^2\right )+b^2 \left (C d^2+e (2 B d+A e)\right )+a \left (a C e^2+2 c d (C d+2 B e)\right )+2 b \left (2 (A c+a C) d e+B \left (c d^2+a e^2\right )\right )\right ) x^5+\frac {1}{6} \left (B \left (c^2 d^2+b^2 e^2+2 c e (2 b d+a e)\right )+2 \left (b^2 C d e+c (A c+2 a C) d e+b \left (c C d^2+A c e^2+a C e^2\right )\right )\right ) x^6+\frac {1}{7} \left (b^2 C e^2+2 c e (2 b C d+b B e+a C e)+c^2 \left (C d^2+e (2 B d+A e)\right )\right ) x^7+\frac {1}{8} c e (2 c C d+B c e+2 b C e) x^8+\frac {1}{9} c^2 C e^2 x^9 \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.00 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=a^2 A d^2 x+\frac {1}{2} a d (a B d+2 A (b d+a e)) x^2+\frac {1}{3} \left (a d (2 b B d+a C d+2 a B e)+A \left (b^2 d^2+4 a b d e+a \left (2 c d^2+a e^2\right )\right )\right ) x^3+\frac {1}{4} \left (b^2 d (B d+2 A e)+a \left (2 B c d^2+4 A c d e+2 a C d e+a B e^2\right )+2 b \left (a d (C d+2 B e)+A \left (c d^2+a e^2\right )\right )\right ) x^4+\frac {1}{5} \left (A c \left (c d^2+2 a e^2\right )+b^2 \left (C d^2+e (2 B d+A e)\right )+a \left (a C e^2+2 c d (C d+2 B e)\right )+2 b \left (2 (A c+a C) d e+B \left (c d^2+a e^2\right )\right )\right ) x^5+\frac {1}{6} \left (B \left (c^2 d^2+b^2 e^2+2 c e (2 b d+a e)\right )+2 \left (b^2 C d e+c (A c+2 a C) d e+b \left (c C d^2+A c e^2+a C e^2\right )\right )\right ) x^6+\frac {1}{7} \left (b^2 C e^2+2 c e (2 b C d+b B e+a C e)+c^2 \left (C d^2+e (2 B d+A e)\right )\right ) x^7+\frac {1}{8} c e (2 c C d+B c e+2 b C e) x^8+\frac {1}{9} c^2 C e^2 x^9 \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.67 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, 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.

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

\(\Big \downarrow \) 2188

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

a^2*A*d^2*x + (a*d*(a*B*d + 2*A*(b*d + a*e))*x^2)/2 + ((a*d*(2*b*B*d + a*C 
*d + 2*a*B*e) + A*(b^2*d^2 + 4*a*b*d*e + a*(2*c*d^2 + a*e^2)))*x^3)/3 + (( 
b^2*d*(B*d + 2*A*e) + a*(2*B*c*d^2 + 4*A*c*d*e + 2*a*C*d*e + a*B*e^2) + 2* 
b*(a*d*(C*d + 2*B*e) + A*(c*d^2 + a*e^2)))*x^4)/4 + ((A*c*(c*d^2 + 2*a*e^2 
) + b^2*(C*d^2 + e*(2*B*d + A*e)) + a*(a*C*e^2 + 2*c*d*(C*d + 2*B*e)) + 2* 
b*(2*(A*c + a*C)*d*e + B*(c*d^2 + a*e^2)))*x^5)/5 + ((B*(c^2*d^2 + b^2*e^2 
 + 2*c*e*(2*b*d + a*e)) + 2*(b^2*C*d*e + c*(A*c + 2*a*C)*d*e + b*(c*C*d^2 
+ A*c*e^2 + a*C*e^2)))*x^6)/6 + ((b^2*C*e^2 + 2*c*e*(2*b*C*d + b*B*e + a*C 
*e) + c^2*(C*d^2 + e*(2*B*d + A*e)))*x^7)/7 + (c*e*(2*c*C*d + B*c*e + 2*b* 
C*e)*x^8)/8 + (c^2*C*e^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 = 0.22 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.89

method	result	size
default	\(\frac {c^{2} C \,e^{2} x^{9}}{9}+\frac {\left (B \,c^{2} e^{2}+2 C \left (e b +c d \right ) c e \right ) x^{8}}{8}+\frac {\left (A \,c^{2} e^{2}+2 B \left (e b +c d \right ) c e +C \left (2 c e \left (a e +b d \right )+\left (e b +c d \right )^{2}\right )\right ) x^{7}}{7}+\frac {\left (2 A \left (e b +c d \right ) c e +B \left (2 c e \left (a e +b d \right )+\left (e b +c d \right )^{2}\right )+C \left (2 a c d e +2 \left (a e +b d \right ) \left (e b +c d \right )\right )\right ) x^{6}}{6}+\frac {\left (A \left (2 c e \left (a e +b d \right )+\left (e b +c d \right )^{2}\right )+B \left (2 a c d e +2 \left (a e +b d \right ) \left (e b +c d \right )\right )+C \left (2 a d \left (e b +c d \right )+\left (a e +b d \right )^{2}\right )\right ) x^{5}}{5}+\frac {\left (A \left (2 a c d e +2 \left (a e +b d \right ) \left (e b +c d \right )\right )+B \left (2 a d \left (e b +c d \right )+\left (a e +b d \right )^{2}\right )+2 C a d \left (a e +b d \right )\right ) x^{4}}{4}+\frac {\left (A \left (2 a d \left (e b +c d \right )+\left (a e +b d \right )^{2}\right )+2 B a d \left (a e +b d \right )+C \,a^{2} d^{2}\right ) x^{3}}{3}+\frac {\left (2 A a d \left (a e +b d \right )+B \,a^{2} d^{2}\right ) x^{2}}{2}+a^{2} A \,d^{2} x\)	\(382\)
norman	\(\frac {c^{2} C \,e^{2} x^{9}}{9}+\left (\frac {1}{8} B \,c^{2} e^{2}+\frac {1}{4} C b c \,e^{2}+\frac {1}{4} C \,c^{2} d e \right ) x^{8}+\left (\frac {1}{7} A \,c^{2} e^{2}+\frac {2}{7} B b c \,e^{2}+\frac {2}{7} B \,c^{2} d e +\frac {2}{7} C a c \,e^{2}+\frac {1}{7} b^{2} C \,e^{2}+\frac {4}{7} C b c d e +\frac {1}{7} C \,c^{2} d^{2}\right ) x^{7}+\left (\frac {1}{3} A b c \,e^{2}+\frac {1}{3} A \,c^{2} d e +\frac {1}{3} B a c \,e^{2}+\frac {1}{6} b^{2} e^{2} B +\frac {2}{3} B b c d e +\frac {1}{6} B \,c^{2} d^{2}+\frac {1}{3} C a b \,e^{2}+\frac {2}{3} C a d e c +\frac {1}{3} b^{2} C d e +\frac {1}{3} C b c \,d^{2}\right ) x^{6}+\left (\frac {2}{5} A a c \,e^{2}+\frac {1}{5} A \,b^{2} e^{2}+\frac {4}{5} A b c d e +\frac {1}{5} A \,c^{2} d^{2}+\frac {2}{5} B a b \,e^{2}+\frac {4}{5} B a c d e +\frac {2}{5} B \,b^{2} d e +\frac {2}{5} B b c \,d^{2}+\frac {1}{5} a^{2} C \,e^{2}+\frac {4}{5} C a b d e +\frac {2}{5} C a c \,d^{2}+\frac {1}{5} C \,d^{2} b^{2}\right ) x^{5}+\left (\frac {1}{2} A a b \,e^{2}+A a c d e +\frac {1}{2} A \,b^{2} d e +\frac {1}{2} A b c \,d^{2}+\frac {1}{4} B \,a^{2} e^{2}+B a b d e +\frac {1}{2} B a c \,d^{2}+\frac {1}{4} B \,b^{2} d^{2}+\frac {1}{2} C \,a^{2} d e +\frac {1}{2} C b \,d^{2} a \right ) x^{4}+\left (\frac {1}{3} A \,a^{2} e^{2}+\frac {4}{3} A a b d e +\frac {2}{3} A \,d^{2} a c +\frac {1}{3} A \,b^{2} d^{2}+\frac {2}{3} B \,a^{2} d e +\frac {2}{3} B a b \,d^{2}+\frac {1}{3} C \,a^{2} d^{2}\right ) x^{3}+\left (A \,a^{2} d e +A a b \,d^{2}+\frac {1}{2} B \,a^{2} d^{2}\right ) x^{2}+a^{2} A \,d^{2} x\)	\(477\)
risch	\(\frac {1}{2} x^{2} B \,a^{2} d^{2}+\frac {4}{7} x^{7} C b c d e +\frac {2}{3} x^{6} B b c d e +\frac {2}{3} x^{6} C a d e c +\frac {1}{8} x^{8} B \,c^{2} e^{2}+\frac {1}{7} x^{7} A \,c^{2} e^{2}+\frac {1}{7} x^{7} b^{2} C \,e^{2}+\frac {1}{7} x^{7} C \,c^{2} d^{2}+\frac {1}{6} x^{6} b^{2} e^{2} B +\frac {1}{6} x^{6} B \,c^{2} d^{2}+\frac {1}{5} x^{5} A \,b^{2} e^{2}+\frac {1}{5} x^{5} A \,c^{2} d^{2}+\frac {1}{5} x^{5} a^{2} C \,e^{2}+\frac {1}{5} x^{5} C \,d^{2} b^{2}+\frac {1}{4} x^{4} B \,a^{2} e^{2}+\frac {1}{4} x^{4} B \,b^{2} d^{2}+\frac {1}{3} x^{3} A \,a^{2} e^{2}+\frac {1}{3} x^{3} A \,b^{2} d^{2}+\frac {1}{3} x^{3} C \,a^{2} d^{2}+\frac {1}{4} x^{8} C \,c^{2} d e +\frac {2}{7} x^{7} B b c \,e^{2}+\frac {2}{7} x^{7} B \,c^{2} d e +\frac {2}{7} x^{7} C a c \,e^{2}+\frac {1}{3} x^{6} A b c \,e^{2}+\frac {1}{3} x^{6} A \,c^{2} d e +\frac {1}{3} x^{6} B a c \,e^{2}+\frac {1}{3} x^{6} C a b \,e^{2}+\frac {1}{3} x^{6} b^{2} C d e +\frac {1}{3} x^{6} C b c \,d^{2}+\frac {2}{5} x^{5} A a c \,e^{2}+\frac {2}{5} x^{5} B a b \,e^{2}+\frac {2}{5} x^{5} B \,b^{2} d e +\frac {2}{5} x^{5} B b c \,d^{2}+\frac {2}{5} x^{5} C a c \,d^{2}+\frac {1}{2} x^{4} A a b \,e^{2}+\frac {1}{2} x^{4} A \,b^{2} d e +\frac {4}{5} x^{5} A b c d e +\frac {4}{5} x^{5} B a c d e +\frac {4}{5} x^{5} C a b d e +x^{4} A a c d e +x^{4} B a b d e +\frac {4}{3} x^{3} A a b d e +a^{2} A \,d^{2} x +\frac {1}{2} x^{4} A b c \,d^{2}+\frac {1}{2} x^{4} B a c \,d^{2}+\frac {1}{2} x^{4} C \,a^{2} d e +\frac {1}{2} x^{4} C b \,d^{2} a +\frac {2}{3} x^{3} A \,d^{2} a c +\frac {2}{3} x^{3} B \,a^{2} d e +\frac {2}{3} x^{3} B a b \,d^{2}+x^{2} A \,a^{2} d e +x^{2} A a b \,d^{2}+\frac {1}{4} x^{8} C b c \,e^{2}+\frac {1}{9} c^{2} C \,e^{2} x^{9}\)	\(598\)
parallelrisch	\(\frac {1}{2} x^{2} B \,a^{2} d^{2}+\frac {4}{7} x^{7} C b c d e +\frac {2}{3} x^{6} B b c d e +\frac {2}{3} x^{6} C a d e c +\frac {1}{8} x^{8} B \,c^{2} e^{2}+\frac {1}{7} x^{7} A \,c^{2} e^{2}+\frac {1}{7} x^{7} b^{2} C \,e^{2}+\frac {1}{7} x^{7} C \,c^{2} d^{2}+\frac {1}{6} x^{6} b^{2} e^{2} B +\frac {1}{6} x^{6} B \,c^{2} d^{2}+\frac {1}{5} x^{5} A \,b^{2} e^{2}+\frac {1}{5} x^{5} A \,c^{2} d^{2}+\frac {1}{5} x^{5} a^{2} C \,e^{2}+\frac {1}{5} x^{5} C \,d^{2} b^{2}+\frac {1}{4} x^{4} B \,a^{2} e^{2}+\frac {1}{4} x^{4} B \,b^{2} d^{2}+\frac {1}{3} x^{3} A \,a^{2} e^{2}+\frac {1}{3} x^{3} A \,b^{2} d^{2}+\frac {1}{3} x^{3} C \,a^{2} d^{2}+\frac {1}{4} x^{8} C \,c^{2} d e +\frac {2}{7} x^{7} B b c \,e^{2}+\frac {2}{7} x^{7} B \,c^{2} d e +\frac {2}{7} x^{7} C a c \,e^{2}+\frac {1}{3} x^{6} A b c \,e^{2}+\frac {1}{3} x^{6} A \,c^{2} d e +\frac {1}{3} x^{6} B a c \,e^{2}+\frac {1}{3} x^{6} C a b \,e^{2}+\frac {1}{3} x^{6} b^{2} C d e +\frac {1}{3} x^{6} C b c \,d^{2}+\frac {2}{5} x^{5} A a c \,e^{2}+\frac {2}{5} x^{5} B a b \,e^{2}+\frac {2}{5} x^{5} B \,b^{2} d e +\frac {2}{5} x^{5} B b c \,d^{2}+\frac {2}{5} x^{5} C a c \,d^{2}+\frac {1}{2} x^{4} A a b \,e^{2}+\frac {1}{2} x^{4} A \,b^{2} d e +\frac {4}{5} x^{5} A b c d e +\frac {4}{5} x^{5} B a c d e +\frac {4}{5} x^{5} C a b d e +x^{4} A a c d e +x^{4} B a b d e +\frac {4}{3} x^{3} A a b d e +a^{2} A \,d^{2} x +\frac {1}{2} x^{4} A b c \,d^{2}+\frac {1}{2} x^{4} B a c \,d^{2}+\frac {1}{2} x^{4} C \,a^{2} d e +\frac {1}{2} x^{4} C b \,d^{2} a +\frac {2}{3} x^{3} A \,d^{2} a c +\frac {2}{3} x^{3} B \,a^{2} d e +\frac {2}{3} x^{3} B a b \,d^{2}+x^{2} A \,a^{2} d e +x^{2} A a b \,d^{2}+\frac {1}{4} x^{8} C b c \,e^{2}+\frac {1}{9} c^{2} C \,e^{2} x^{9}\)	\(598\)
gosper	\(\frac {x \left (280 C \,c^{2} e^{2} x^{8}+315 x^{7} B \,c^{2} e^{2}+630 x^{7} C b c \,e^{2}+630 x^{7} C \,c^{2} d e +360 x^{6} A \,c^{2} e^{2}+720 x^{6} B b c \,e^{2}+720 x^{6} B \,c^{2} d e +720 x^{6} C a c \,e^{2}+360 x^{6} b^{2} C \,e^{2}+1440 x^{6} C b c d e +360 x^{6} C \,c^{2} d^{2}+840 x^{5} A b c \,e^{2}+840 x^{5} A \,c^{2} d e +840 x^{5} B a c \,e^{2}+420 x^{5} b^{2} e^{2} B +1680 x^{5} B b c d e +420 x^{5} B \,c^{2} d^{2}+840 x^{5} C a b \,e^{2}+1680 x^{5} C a d e c +840 x^{5} b^{2} C d e +840 x^{5} C b c \,d^{2}+1008 x^{4} A a c \,e^{2}+504 x^{4} A \,b^{2} e^{2}+2016 x^{4} A b c d e +504 x^{4} A \,c^{2} d^{2}+1008 x^{4} B a b \,e^{2}+2016 x^{4} B a c d e +1008 x^{4} B \,b^{2} d e +1008 x^{4} B b c \,d^{2}+504 x^{4} a^{2} C \,e^{2}+2016 x^{4} C a b d e +1008 x^{4} C a c \,d^{2}+504 x^{4} C \,d^{2} b^{2}+1260 x^{3} A a b \,e^{2}+2520 x^{3} A a c d e +1260 x^{3} A \,b^{2} d e +1260 x^{3} A b c \,d^{2}+630 x^{3} B \,a^{2} e^{2}+2520 x^{3} B a b d e +1260 x^{3} B a c \,d^{2}+630 x^{3} B \,b^{2} d^{2}+1260 x^{3} C \,a^{2} d e +1260 x^{3} C b \,d^{2} a +840 x^{2} A \,a^{2} e^{2}+3360 x^{2} A a b d e +1680 x^{2} A \,d^{2} a c +840 x^{2} A \,b^{2} d^{2}+1680 x^{2} B \,a^{2} d e +1680 x^{2} B a b \,d^{2}+840 x^{2} C \,a^{2} d^{2}+2520 x A \,a^{2} d e +2520 x A a b \,d^{2}+1260 x B \,a^{2} d^{2}+2520 A \,a^{2} d^{2}\right )}{2520}\)	\(599\)
orering	\(\frac {x \left (280 C \,c^{2} e^{2} x^{8}+315 x^{7} B \,c^{2} e^{2}+630 x^{7} C b c \,e^{2}+630 x^{7} C \,c^{2} d e +360 x^{6} A \,c^{2} e^{2}+720 x^{6} B b c \,e^{2}+720 x^{6} B \,c^{2} d e +720 x^{6} C a c \,e^{2}+360 x^{6} b^{2} C \,e^{2}+1440 x^{6} C b c d e +360 x^{6} C \,c^{2} d^{2}+840 x^{5} A b c \,e^{2}+840 x^{5} A \,c^{2} d e +840 x^{5} B a c \,e^{2}+420 x^{5} b^{2} e^{2} B +1680 x^{5} B b c d e +420 x^{5} B \,c^{2} d^{2}+840 x^{5} C a b \,e^{2}+1680 x^{5} C a d e c +840 x^{5} b^{2} C d e +840 x^{5} C b c \,d^{2}+1008 x^{4} A a c \,e^{2}+504 x^{4} A \,b^{2} e^{2}+2016 x^{4} A b c d e +504 x^{4} A \,c^{2} d^{2}+1008 x^{4} B a b \,e^{2}+2016 x^{4} B a c d e +1008 x^{4} B \,b^{2} d e +1008 x^{4} B b c \,d^{2}+504 x^{4} a^{2} C \,e^{2}+2016 x^{4} C a b d e +1008 x^{4} C a c \,d^{2}+504 x^{4} C \,d^{2} b^{2}+1260 x^{3} A a b \,e^{2}+2520 x^{3} A a c d e +1260 x^{3} A \,b^{2} d e +1260 x^{3} A b c \,d^{2}+630 x^{3} B \,a^{2} e^{2}+2520 x^{3} B a b d e +1260 x^{3} B a c \,d^{2}+630 x^{3} B \,b^{2} d^{2}+1260 x^{3} C \,a^{2} d e +1260 x^{3} C b \,d^{2} a +840 x^{2} A \,a^{2} e^{2}+3360 x^{2} A a b d e +1680 x^{2} A \,d^{2} a c +840 x^{2} A \,b^{2} d^{2}+1680 x^{2} B \,a^{2} d e +1680 x^{2} B a b \,d^{2}+840 x^{2} C \,a^{2} d^{2}+2520 x A \,a^{2} d e +2520 x A a b \,d^{2}+1260 x B \,a^{2} d^{2}+2520 A \,a^{2} d^{2}\right ) \left (a d +\left (a e +b d \right ) x +\left (e b +c d \right ) x^{2}+c e \,x^{3}\right )^{2}}{2520 \left (c \,x^{2}+b x +a \right )^{2} \left (e x +d \right )^{2}}\)	\(650\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.95 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=\frac {1}{9} \, C c^{2} e^{2} x^{9} + \frac {1}{8} \, {\left (2 \, C c^{2} d e + {\left (2 \, C b c + B c^{2}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (C c^{2} d^{2} + 2 \, {\left (2 \, C b c + B c^{2}\right )} d e + {\left (C b^{2} + A c^{2} + 2 \, {\left (C a + B b\right )} c\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left ({\left (2 \, C b c + B c^{2}\right )} d^{2} + 2 \, {\left (C b^{2} + A c^{2} + 2 \, {\left (C a + B b\right )} c\right )} d e + {\left (2 \, C a b + B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{2}\right )} x^{6} + A a^{2} d^{2} x + \frac {1}{5} \, {\left ({\left (C b^{2} + A c^{2} + 2 \, {\left (C a + B b\right )} c\right )} d^{2} + 2 \, {\left (2 \, C a b + B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e + {\left (C a^{2} + 2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left ({\left (2 \, C a b + B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} + 2 \, {\left (C a^{2} + 2 \, B a b + A b^{2} + 2 \, A a c\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{2} e^{2} + {\left (C a^{2} + 2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{2} d e + {\left (B a^{2} + 2 \, A a b\right )} d^{2}\right )} x^{2} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.39 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=A a^{2} d^{2} x + \frac {C c^{2} e^{2} x^{9}}{9} + x^{8} \left (\frac {B c^{2} e^{2}}{8} + \frac {C b c e^{2}}{4} + \frac {C c^{2} d e}{4}\right ) + x^{7} \left (\frac {A c^{2} e^{2}}{7} + \frac {2 B b c e^{2}}{7} + \frac {2 B c^{2} d e}{7} + \frac {2 C a c e^{2}}{7} + \frac {C b^{2} e^{2}}{7} + \frac {4 C b c d e}{7} + \frac {C c^{2} d^{2}}{7}\right ) + x^{6} \left (\frac {A b c e^{2}}{3} + \frac {A c^{2} d e}{3} + \frac {B a c e^{2}}{3} + \frac {B b^{2} e^{2}}{6} + \frac {2 B b c d e}{3} + \frac {B c^{2} d^{2}}{6} + \frac {C a b e^{2}}{3} + \frac {2 C a c d e}{3} + \frac {C b^{2} d e}{3} + \frac {C b c d^{2}}{3}\right ) + x^{5} \cdot \left (\frac {2 A a c e^{2}}{5} + \frac {A b^{2} e^{2}}{5} + \frac {4 A b c d e}{5} + \frac {A c^{2} d^{2}}{5} + \frac {2 B a b e^{2}}{5} + \frac {4 B a c d e}{5} + \frac {2 B b^{2} d e}{5} + \frac {2 B b c d^{2}}{5} + \frac {C a^{2} e^{2}}{5} + \frac {4 C a b d e}{5} + \frac {2 C a c d^{2}}{5} + \frac {C b^{2} d^{2}}{5}\right ) + x^{4} \left (\frac {A a b e^{2}}{2} + A a c d e + \frac {A b^{2} d e}{2} + \frac {A b c d^{2}}{2} + \frac {B a^{2} e^{2}}{4} + B a b d e + \frac {B a c d^{2}}{2} + \frac {B b^{2} d^{2}}{4} + \frac {C a^{2} d e}{2} + \frac {C a b d^{2}}{2}\right ) + x^{3} \left (\frac {A a^{2} e^{2}}{3} + \frac {4 A a b d e}{3} + \frac {2 A a c d^{2}}{3} + \frac {A b^{2} d^{2}}{3} + \frac {2 B a^{2} d e}{3} + \frac {2 B a b d^{2}}{3} + \frac {C a^{2} d^{2}}{3}\right ) + x^{2} \left (A a^{2} d e + A a b d^{2} + \frac {B a^{2} d^{2}}{2}\right ) \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.95 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=\frac {1}{9} \, C c^{2} e^{2} x^{9} + \frac {1}{8} \, {\left (2 \, C c^{2} d e + {\left (2 \, C b c + B c^{2}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (C c^{2} d^{2} + 2 \, {\left (2 \, C b c + B c^{2}\right )} d e + {\left (C b^{2} + A c^{2} + 2 \, {\left (C a + B b\right )} c\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left ({\left (2 \, C b c + B c^{2}\right )} d^{2} + 2 \, {\left (C b^{2} + A c^{2} + 2 \, {\left (C a + B b\right )} c\right )} d e + {\left (2 \, C a b + B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{2}\right )} x^{6} + A a^{2} d^{2} x + \frac {1}{5} \, {\left ({\left (C b^{2} + A c^{2} + 2 \, {\left (C a + B b\right )} c\right )} d^{2} + 2 \, {\left (2 \, C a b + B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e + {\left (C a^{2} + 2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left ({\left (2 \, C a b + B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} + 2 \, {\left (C a^{2} + 2 \, B a b + A b^{2} + 2 \, A a c\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{2} e^{2} + {\left (C a^{2} + 2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{2} d e + {\left (B a^{2} + 2 \, A a b\right )} d^{2}\right )} x^{2} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.39 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=\frac {1}{9} \, C c^{2} e^{2} x^{9} + \frac {1}{4} \, C c^{2} d e x^{8} + \frac {1}{4} \, C b c e^{2} x^{8} + \frac {1}{8} \, B c^{2} e^{2} x^{8} + \frac {1}{7} \, C c^{2} d^{2} x^{7} + \frac {4}{7} \, C b c d e x^{7} + \frac {2}{7} \, B c^{2} d e x^{7} + \frac {1}{7} \, C b^{2} e^{2} x^{7} + \frac {2}{7} \, C a c e^{2} x^{7} + \frac {2}{7} \, B b c e^{2} x^{7} + \frac {1}{7} \, A c^{2} e^{2} x^{7} + \frac {1}{3} \, C b c d^{2} x^{6} + \frac {1}{6} \, B c^{2} d^{2} x^{6} + \frac {1}{3} \, C b^{2} d e x^{6} + \frac {2}{3} \, C a c d e x^{6} + \frac {2}{3} \, B b c d e x^{6} + \frac {1}{3} \, A c^{2} d e x^{6} + \frac {1}{3} \, C a b e^{2} x^{6} + \frac {1}{6} \, B b^{2} e^{2} x^{6} + \frac {1}{3} \, B a c e^{2} x^{6} + \frac {1}{3} \, A b c e^{2} x^{6} + \frac {1}{5} \, C b^{2} d^{2} x^{5} + \frac {2}{5} \, C a c d^{2} x^{5} + \frac {2}{5} \, B b c d^{2} x^{5} + \frac {1}{5} \, A c^{2} d^{2} x^{5} + \frac {4}{5} \, C a b d e x^{5} + \frac {2}{5} \, B b^{2} d e x^{5} + \frac {4}{5} \, B a c d e x^{5} + \frac {4}{5} \, A b c d e x^{5} + \frac {1}{5} \, C a^{2} e^{2} x^{5} + \frac {2}{5} \, B a b e^{2} x^{5} + \frac {1}{5} \, A b^{2} e^{2} x^{5} + \frac {2}{5} \, A a c e^{2} x^{5} + \frac {1}{2} \, C a b d^{2} x^{4} + \frac {1}{4} \, B b^{2} d^{2} x^{4} + \frac {1}{2} \, B a c d^{2} x^{4} + \frac {1}{2} \, A b c d^{2} x^{4} + \frac {1}{2} \, C a^{2} d e x^{4} + B a b d e x^{4} + \frac {1}{2} \, A b^{2} d e x^{4} + A a c d e x^{4} + \frac {1}{4} \, B a^{2} e^{2} x^{4} + \frac {1}{2} \, A a b e^{2} x^{4} + \frac {1}{3} \, C a^{2} d^{2} x^{3} + \frac {2}{3} \, B a b d^{2} x^{3} + \frac {1}{3} \, A b^{2} d^{2} x^{3} + \frac {2}{3} \, A a c d^{2} x^{3} + \frac {2}{3} \, B a^{2} d e x^{3} + \frac {4}{3} \, A a b d e x^{3} + \frac {1}{3} \, A a^{2} e^{2} x^{3} + \frac {1}{2} \, B a^{2} d^{2} x^{2} + A a b d^{2} x^{2} + A a^{2} d e x^{2} + A a^{2} d^{2} x \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.07 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=x^4\,\left (\frac {C\,a^2\,d\,e}{2}+\frac {B\,a^2\,e^2}{4}+\frac {C\,a\,b\,d^2}{2}+B\,a\,b\,d\,e+\frac {A\,a\,b\,e^2}{2}+\frac {B\,c\,a\,d^2}{2}+A\,c\,a\,d\,e+\frac {B\,b^2\,d^2}{4}+\frac {A\,b^2\,d\,e}{2}+\frac {A\,c\,b\,d^2}{2}\right )+x^6\,\left (\frac {C\,b^2\,d\,e}{3}+\frac {B\,b^2\,e^2}{6}+\frac {C\,b\,c\,d^2}{3}+\frac {2\,B\,b\,c\,d\,e}{3}+\frac {A\,b\,c\,e^2}{3}+\frac {C\,a\,b\,e^2}{3}+\frac {B\,c^2\,d^2}{6}+\frac {A\,c^2\,d\,e}{3}+\frac {2\,C\,a\,c\,d\,e}{3}+\frac {B\,a\,c\,e^2}{3}\right )+x^5\,\left (\frac {C\,a^2\,e^2}{5}+\frac {4\,C\,a\,b\,d\,e}{5}+\frac {2\,B\,a\,b\,e^2}{5}+\frac {2\,C\,a\,c\,d^2}{5}+\frac {4\,B\,a\,c\,d\,e}{5}+\frac {2\,A\,a\,c\,e^2}{5}+\frac {C\,b^2\,d^2}{5}+\frac {2\,B\,b^2\,d\,e}{5}+\frac {A\,b^2\,e^2}{5}+\frac {2\,B\,b\,c\,d^2}{5}+\frac {4\,A\,b\,c\,d\,e}{5}+\frac {A\,c^2\,d^2}{5}\right )+x^3\,\left (\frac {C\,a^2\,d^2}{3}+\frac {2\,B\,a^2\,d\,e}{3}+\frac {A\,a^2\,e^2}{3}+\frac {2\,B\,a\,b\,d^2}{3}+\frac {4\,A\,a\,b\,d\,e}{3}+\frac {2\,A\,c\,a\,d^2}{3}+\frac {A\,b^2\,d^2}{3}\right )+x^7\,\left (\frac {C\,b^2\,e^2}{7}+\frac {4\,C\,b\,c\,d\,e}{7}+\frac {2\,B\,b\,c\,e^2}{7}+\frac {C\,c^2\,d^2}{7}+\frac {2\,B\,c^2\,d\,e}{7}+\frac {A\,c^2\,e^2}{7}+\frac {2\,C\,a\,c\,e^2}{7}\right )+\frac {C\,c^2\,e^2\,x^9}{9}+\frac {a\,d\,x^2\,\left (2\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {c\,e\,x^8\,\left (B\,c\,e+2\,C\,b\,e+2\,C\,c\,d\right )}{8}+A\,a^2\,d^2\,x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.78 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^2 \, dx=\frac {x \left (280 c^{3} e^{2} x^{8}+945 b \,c^{2} e^{2} x^{7}+630 c^{3} d e \,x^{7}+1080 a \,c^{2} e^{2} x^{6}+1080 b^{2} c \,e^{2} x^{6}+2160 b \,c^{2} d e \,x^{6}+360 c^{3} d^{2} x^{6}+2520 a b c \,e^{2} x^{5}+2520 a \,c^{2} d e \,x^{5}+420 b^{3} e^{2} x^{5}+2520 b^{2} c d e \,x^{5}+1260 b \,c^{2} d^{2} x^{5}+1512 a^{2} c \,e^{2} x^{4}+1512 a \,b^{2} e^{2} x^{4}+6048 a b c d e \,x^{4}+1512 a \,c^{2} d^{2} x^{4}+1008 b^{3} d e \,x^{4}+1512 b^{2} c \,d^{2} x^{4}+1890 a^{2} b \,e^{2} x^{3}+3780 a^{2} c d e \,x^{3}+3780 a \,b^{2} d e \,x^{3}+3780 a b c \,d^{2} x^{3}+630 b^{3} d^{2} x^{3}+840 a^{3} e^{2} x^{2}+5040 a^{2} b d e \,x^{2}+2520 a^{2} c \,d^{2} x^{2}+2520 a \,b^{2} d^{2} x^{2}+2520 a^{3} d e x +3780 a^{2} b \,d^{2} x +2520 a^{3} d^{2}\right )}{2520} \] Input:

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

Output:

(x*(2520*a**3*d**2 + 2520*a**3*d*e*x + 840*a**3*e**2*x**2 + 3780*a**2*b*d* 
*2*x + 5040*a**2*b*d*e*x**2 + 1890*a**2*b*e**2*x**3 + 2520*a**2*c*d**2*x** 
2 + 3780*a**2*c*d*e*x**3 + 1512*a**2*c*e**2*x**4 + 2520*a*b**2*d**2*x**2 + 
 3780*a*b**2*d*e*x**3 + 1512*a*b**2*e**2*x**4 + 3780*a*b*c*d**2*x**3 + 604 
8*a*b*c*d*e*x**4 + 2520*a*b*c*e**2*x**5 + 1512*a*c**2*d**2*x**4 + 2520*a*c 
**2*d*e*x**5 + 1080*a*c**2*e**2*x**6 + 630*b**3*d**2*x**3 + 1008*b**3*d*e* 
x**4 + 420*b**3*e**2*x**5 + 1512*b**2*c*d**2*x**4 + 2520*b**2*c*d*e*x**5 + 
 1080*b**2*c*e**2*x**6 + 1260*b*c**2*d**2*x**5 + 2160*b*c**2*d*e*x**6 + 94 
5*b*c**2*e**2*x**7 + 360*c**3*d**2*x**6 + 630*c**3*d*e*x**7 + 280*c**3*e** 
2*x**8))/2520