\(\int (A+B x+C x^2) (3 a b+3 b^2 x+3 b c x^2+c^2 x^3)^3 \, dx\) [68]

Optimal result

Integrand size = 38, antiderivative size = 310 \[ \int \left (A+B x+C x^2\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=\frac {b^3 \left (b^2-3 a c\right )^3 \left (b B c-A c^2-b^2 C\right ) x}{c^5}-\frac {b^3 \left (b^2-3 a c\right )^3 (B c-2 b C) (b+c x)^2}{2 c^6}-\frac {3 b^2 \left (b^2-3 a c\right )^2 \left (b B c-A c^2-b^2 C\right ) (b+c x)^4}{4 c^6}+\frac {3 b^2 \left (b^2-3 a c\right )^2 (B c-2 b C) (b+c x)^5}{5 c^6}+\frac {3 b \left (b^2-3 a c\right ) \left (b B c-A c^2-b^2 C\right ) (b+c x)^7}{7 c^6}-\frac {3 b \left (b^2-3 a c\right ) (B c-2 b C) (b+c x)^8}{8 c^6}-\frac {\left (b B c-A c^2-b^2 C\right ) (b+c x)^{10}}{10 c^6}+\frac {(B c-2 b C) (b+c x)^{11}}{11 c^6}+\frac {C \left (b \left (3 a-\frac {b^2}{c}\right )+\frac {(b+c x)^3}{c}\right )^4}{12 c^2} \] Output:

b^3*(-3*a*c+b^2)^3*(-A*c^2+B*b*c-C*b^2)*x/c^5-1/2*b^3*(-3*a*c+b^2)^3*(B*c- 
2*C*b)*(c*x+b)^2/c^6-3/4*b^2*(-3*a*c+b^2)^2*(-A*c^2+B*b*c-C*b^2)*(c*x+b)^4 
/c^6+3/5*b^2*(-3*a*c+b^2)^2*(B*c-2*C*b)*(c*x+b)^5/c^6+3/7*b*(-3*a*c+b^2)*( 
-A*c^2+B*b*c-C*b^2)*(c*x+b)^7/c^6-3/8*b*(-3*a*c+b^2)*(B*c-2*C*b)*(c*x+b)^8 
/c^6-1/10*(-A*c^2+B*b*c-C*b^2)*(c*x+b)^10/c^6+1/11*(B*c-2*C*b)*(c*x+b)^11/ 
c^6+1/12*C*(b*(3*a-b^2/c)+(c*x+b)^3/c)^4/c^2
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.32 \[ \int \left (A+B x+C x^2\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=27 a^3 A b^3 x+\frac {27}{2} a^2 b^3 (3 A b+a B) x^2+9 a b^3 \left (3 A \left (b^2+a c\right )+a (3 b B+a C)\right ) x^3+\frac {27}{4} b^2 \left (A \left (b^4+6 a b^2 c+a^2 c^2\right )+3 a b \left (b^2 B+a B c+a b C\right )\right ) x^4+\frac {27}{5} b^2 \left (b^4 B+6 a b^2 B c+a^2 B c^2+3 b^3 (A c+a C)+a b c (5 A c+3 a C)\right ) x^5+\frac {9}{2} b^2 \left (3 b^3 B c+5 a b B c^2+b^4 C+a c^2 (2 A c+a C)+2 b^2 c (2 A c+3 a C)\right ) x^6+\frac {9}{7} b c \left (12 b^3 B c+6 a b B c^2+a A c^3+9 b^4 C+3 b^2 c (3 A c+5 a C)\right ) x^7+\frac {9}{8} b c^2 \left (9 b^2 B c+a B c^2+12 b^3 C+2 b c (2 A c+3 a C)\right ) x^8+b c^3 \left (4 b B c+9 b^2 C+c (A c+a C)\right ) x^9+\frac {1}{10} c^4 \left (9 b B c+A c^2+36 b^2 C\right ) x^{10}+\frac {1}{11} c^5 (B c+9 b C) x^{11}+\frac {1}{12} c^6 C x^{12} \] Input:

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

Output:

27*a^3*A*b^3*x + (27*a^2*b^3*(3*A*b + a*B)*x^2)/2 + 9*a*b^3*(3*A*(b^2 + a* 
c) + a*(3*b*B + a*C))*x^3 + (27*b^2*(A*(b^4 + 6*a*b^2*c + a^2*c^2) + 3*a*b 
*(b^2*B + a*B*c + a*b*C))*x^4)/4 + (27*b^2*(b^4*B + 6*a*b^2*B*c + a^2*B*c^ 
2 + 3*b^3*(A*c + a*C) + a*b*c*(5*A*c + 3*a*C))*x^5)/5 + (9*b^2*(3*b^3*B*c 
+ 5*a*b*B*c^2 + b^4*C + a*c^2*(2*A*c + a*C) + 2*b^2*c*(2*A*c + 3*a*C))*x^6 
)/2 + (9*b*c*(12*b^3*B*c + 6*a*b*B*c^2 + a*A*c^3 + 9*b^4*C + 3*b^2*c*(3*A* 
c + 5*a*C))*x^7)/7 + (9*b*c^2*(9*b^2*B*c + a*B*c^2 + 12*b^3*C + 2*b*c*(2*A 
*c + 3*a*C))*x^8)/8 + b*c^3*(4*b*B*c + 9*b^2*C + c*(A*c + a*C))*x^9 + (c^4 
*(9*b*B*c + A*c^2 + 36*b^2*C)*x^10)/10 + (c^5*(B*c + 9*b*C)*x^11)/11 + (c^ 
6*C*x^12)/12
 

Rubi [A] (verified)

Time = 1.49 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.32, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, 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.

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

\(\Big \downarrow \) 2188

\(\displaystyle \int \left (27 a^3 A b^3+27 a^2 b^3 x (a B+3 A b)+27 b^2 x^3 \left (A \left (a^2 c^2+6 a b^2 c+b^4\right )+3 a b \left (a b C+a B c+b^2 B\right )\right )+27 b^2 x^4 \left (a^2 B c^2+3 b^3 (a C+A c)+a b c (3 a C+5 A c)+6 a b^2 B c+b^4 B\right )+9 b c^3 x^8 \left (c (a C+A c)+9 b^2 C+4 b B c\right )+9 b c^2 x^7 \left (2 b c (3 a C+2 A c)+a B c^2+12 b^3 C+9 b^2 B c\right )+27 a b^3 x^2 \left (3 A \left (a c+b^2\right )+a (a C+3 b B)\right )+27 b^2 x^5 \left (2 b^2 c (3 a C+2 A c)+a c^2 (a C+2 A c)+5 a b B c^2+b^4 C+3 b^3 B c\right )+9 b c x^6 \left (3 b^2 c (5 a C+3 A c)+a A c^3+6 a b B c^2+9 b^4 C+12 b^3 B c\right )+c^4 x^9 \left (A c^2+36 b^2 C+9 b B c\right )+c^5 x^{10} (9 b C+B c)+c^6 C x^{11}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 27 a^3 A b^3 x+\frac {27}{2} a^2 b^3 x^2 (a B+3 A b)+\frac {27}{4} b^2 x^4 \left (A \left (a^2 c^2+6 a b^2 c+b^4\right )+3 a b \left (a b C+a B c+b^2 B\right )\right )+\frac {27}{5} b^2 x^5 \left (a^2 B c^2+3 b^3 (a C+A c)+a b c (3 a C+5 A c)+6 a b^2 B c+b^4 B\right )+b c^3 x^9 \left (c (a C+A c)+9 b^2 C+4 b B c\right )+\frac {9}{8} b c^2 x^8 \left (2 b c (3 a C+2 A c)+a B c^2+12 b^3 C+9 b^2 B c\right )+9 a b^3 x^3 \left (3 A \left (a c+b^2\right )+a (a C+3 b B)\right )+\frac {9}{2} b^2 x^6 \left (2 b^2 c (3 a C+2 A c)+a c^2 (a C+2 A c)+5 a b B c^2+b^4 C+3 b^3 B c\right )+\frac {9}{7} b c x^7 \left (3 b^2 c (5 a C+3 A c)+a A c^3+6 a b B c^2+9 b^4 C+12 b^3 B c\right )+\frac {1}{10} c^4 x^{10} \left (A c^2+36 b^2 C+9 b B c\right )+\frac {1}{11} c^5 x^{11} (9 b C+B c)+\frac {1}{12} c^6 C x^{12}\)

Input:

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

Output:

27*a^3*A*b^3*x + (27*a^2*b^3*(3*A*b + a*B)*x^2)/2 + 9*a*b^3*(3*A*(b^2 + a* 
c) + a*(3*b*B + a*C))*x^3 + (27*b^2*(A*(b^4 + 6*a*b^2*c + a^2*c^2) + 3*a*b 
*(b^2*B + a*B*c + a*b*C))*x^4)/4 + (27*b^2*(b^4*B + 6*a*b^2*B*c + a^2*B*c^ 
2 + 3*b^3*(A*c + a*C) + a*b*c*(5*A*c + 3*a*C))*x^5)/5 + (9*b^2*(3*b^3*B*c 
+ 5*a*b*B*c^2 + b^4*C + a*c^2*(2*A*c + a*C) + 2*b^2*c*(2*A*c + 3*a*C))*x^6 
)/2 + (9*b*c*(12*b^3*B*c + 6*a*b*B*c^2 + a*A*c^3 + 9*b^4*C + 3*b^2*c*(3*A* 
c + 5*a*C))*x^7)/7 + (9*b*c^2*(9*b^2*B*c + a*B*c^2 + 12*b^3*C + 2*b*c*(2*A 
*c + 3*a*C))*x^8)/8 + b*c^3*(4*b*B*c + 9*b^2*C + c*(A*c + a*C))*x^9 + (c^4 
*(9*b*B*c + A*c^2 + 36*b^2*C)*x^10)/10 + (c^5*(B*c + 9*b*C)*x^11)/11 + (c^ 
6*C*x^12)/12
 

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 = 0.19 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.49

method	result	size
norman	\(\frac {C \,c^{6} x^{12}}{12}+\left (\frac {1}{11} B \,c^{6}+\frac {9}{11} C b \,c^{5}\right ) x^{11}+\left (\frac {1}{10} A \,c^{6}+\frac {9}{10} B b \,c^{5}+\frac {18}{5} C \,b^{2} c^{4}\right ) x^{10}+\left (A b \,c^{5}+4 B \,b^{2} c^{4}+C a b \,c^{4}+9 C \,b^{3} c^{3}\right ) x^{9}+\left (\frac {9}{2} A \,b^{2} c^{4}+\frac {9}{8} B a b \,c^{4}+\frac {81}{8} B \,b^{3} c^{3}+\frac {27}{4} C a \,b^{2} c^{3}+\frac {27}{2} C \,b^{4} c^{2}\right ) x^{8}+\left (\frac {9}{7} A a b \,c^{4}+\frac {81}{7} A \,b^{3} c^{3}+\frac {54}{7} B a \,b^{2} c^{3}+\frac {108}{7} B \,b^{4} c^{2}+\frac {135}{7} C a \,b^{3} c^{2}+\frac {81}{7} C \,b^{5} c \right ) x^{7}+\left (9 A a \,b^{2} c^{3}+18 A \,b^{4} c^{2}+\frac {45}{2} B a \,b^{3} c^{2}+\frac {27}{2} B \,b^{5} c +\frac {9}{2} C \,a^{2} b^{2} c^{2}+27 C a \,b^{4} c +\frac {9}{2} C \,b^{6}\right ) x^{6}+\left (27 A a \,b^{3} c^{2}+\frac {81}{5} A \,b^{5} c +\frac {27}{5} B \,a^{2} b^{2} c^{2}+\frac {162}{5} B a \,b^{4} c +\frac {27}{5} B \,b^{6}+\frac {81}{5} C \,a^{2} b^{3} c +\frac {81}{5} C \,b^{5} a \right ) x^{5}+\left (\frac {27}{4} A \,a^{2} b^{2} c^{2}+\frac {81}{2} A a \,b^{4} c +\frac {27}{4} A \,b^{6}+\frac {81}{4} B \,a^{2} b^{3} c +\frac {81}{4} B \,b^{5} a +\frac {81}{4} b^{4} C \,a^{2}\right ) x^{4}+\left (27 A \,a^{2} b^{3} c +27 A \,b^{5} a +27 B \,b^{4} a^{2}+9 C \,b^{3} a^{3}\right ) x^{3}+\left (\frac {81}{2} A \,a^{2} b^{4}+\frac {27}{2} B \,a^{3} b^{3}\right ) x^{2}+27 A x \,a^{3} b^{3}\)	\(462\)
gosper	\(\frac {27}{2} B \,x^{2} a^{3} b^{3}+27 A x \,a^{3} b^{3}+\frac {81}{4} B \,x^{4} a \,b^{5}+\frac {81}{4} C \,x^{4} a^{2} b^{4}+27 A \,a^{2} b^{3} c \,x^{3}+\frac {27}{4} x^{4} A \,a^{2} b^{2} c^{2}+\frac {81}{2} x^{4} A a \,b^{4} c +\frac {81}{4} x^{4} B \,a^{2} b^{3} c +C a b \,c^{4} x^{9}+\frac {27}{5} x^{5} B \,a^{2} b^{2} c^{2}+\frac {162}{5} x^{5} B a \,b^{4} c +\frac {81}{5} x^{5} C \,a^{2} b^{3} c +\frac {9}{2} x^{6} C \,a^{2} b^{2} c^{2}+27 x^{6} C a \,b^{4} c +27 x^{5} A a \,b^{3} c^{2}+9 x^{6} A a \,b^{2} c^{3}+\frac {45}{2} x^{6} B a \,b^{3} c^{2}+\frac {54}{7} x^{7} B a \,b^{2} c^{3}+\frac {135}{7} x^{7} C a \,b^{3} c^{2}+\frac {9}{7} x^{7} A a b \,c^{4}+\frac {9}{8} x^{8} B a b \,c^{4}+\frac {27}{4} x^{8} C a \,b^{2} c^{3}+4 B \,b^{2} c^{4} x^{9}+9 C \,b^{3} c^{3} x^{9}+A b \,c^{5} x^{9}+18 x^{6} A \,b^{4} c^{2}+\frac {27}{2} x^{6} B \,b^{5} c +\frac {81}{5} x^{5} A \,b^{5} c +\frac {108}{7} x^{7} B \,b^{4} c^{2}+\frac {81}{7} x^{7} C \,b^{5} c +\frac {81}{7} x^{7} A \,b^{3} c^{3}+\frac {27}{2} x^{8} C \,b^{4} c^{2}+\frac {9}{11} x^{11} C b \,c^{5}+\frac {9}{10} x^{10} B b \,c^{5}+\frac {18}{5} x^{10} C \,b^{2} c^{4}+\frac {9}{2} x^{8} A \,b^{2} c^{4}+\frac {81}{8} x^{8} B \,b^{3} c^{3}+\frac {81}{5} C \,b^{5} a \,x^{5}+27 A \,x^{3} a \,b^{5}+\frac {81}{2} A \,x^{2} a^{2} b^{4}+27 B \,x^{3} a^{2} b^{4}+9 C \,x^{3} a^{3} b^{3}+\frac {1}{12} C \,c^{6} x^{12}+\frac {1}{11} x^{11} B \,c^{6}+\frac {1}{10} x^{10} A \,c^{6}+\frac {27}{5} x^{5} B \,b^{6}+\frac {27}{4} A \,x^{4} b^{6}+\frac {9}{2} x^{6} C \,b^{6}\)	\(550\)
risch	\(\frac {27}{2} B \,x^{2} a^{3} b^{3}+27 A x \,a^{3} b^{3}+\frac {81}{4} B \,x^{4} a \,b^{5}+\frac {81}{4} C \,x^{4} a^{2} b^{4}+27 A \,a^{2} b^{3} c \,x^{3}+\frac {27}{4} x^{4} A \,a^{2} b^{2} c^{2}+\frac {81}{2} x^{4} A a \,b^{4} c +\frac {81}{4} x^{4} B \,a^{2} b^{3} c +C a b \,c^{4} x^{9}+\frac {27}{5} x^{5} B \,a^{2} b^{2} c^{2}+\frac {162}{5} x^{5} B a \,b^{4} c +\frac {81}{5} x^{5} C \,a^{2} b^{3} c +\frac {9}{2} x^{6} C \,a^{2} b^{2} c^{2}+27 x^{6} C a \,b^{4} c +27 x^{5} A a \,b^{3} c^{2}+9 x^{6} A a \,b^{2} c^{3}+\frac {45}{2} x^{6} B a \,b^{3} c^{2}+\frac {54}{7} x^{7} B a \,b^{2} c^{3}+\frac {135}{7} x^{7} C a \,b^{3} c^{2}+\frac {9}{7} x^{7} A a b \,c^{4}+\frac {9}{8} x^{8} B a b \,c^{4}+\frac {27}{4} x^{8} C a \,b^{2} c^{3}+4 B \,b^{2} c^{4} x^{9}+9 C \,b^{3} c^{3} x^{9}+A b \,c^{5} x^{9}+18 x^{6} A \,b^{4} c^{2}+\frac {27}{2} x^{6} B \,b^{5} c +\frac {81}{5} x^{5} A \,b^{5} c +\frac {108}{7} x^{7} B \,b^{4} c^{2}+\frac {81}{7} x^{7} C \,b^{5} c +\frac {81}{7} x^{7} A \,b^{3} c^{3}+\frac {27}{2} x^{8} C \,b^{4} c^{2}+\frac {9}{11} x^{11} C b \,c^{5}+\frac {9}{10} x^{10} B b \,c^{5}+\frac {18}{5} x^{10} C \,b^{2} c^{4}+\frac {9}{2} x^{8} A \,b^{2} c^{4}+\frac {81}{8} x^{8} B \,b^{3} c^{3}+\frac {81}{5} C \,b^{5} a \,x^{5}+27 A \,x^{3} a \,b^{5}+\frac {81}{2} A \,x^{2} a^{2} b^{4}+27 B \,x^{3} a^{2} b^{4}+9 C \,x^{3} a^{3} b^{3}+\frac {1}{12} C \,c^{6} x^{12}+\frac {1}{11} x^{11} B \,c^{6}+\frac {1}{10} x^{10} A \,c^{6}+\frac {27}{5} x^{5} B \,b^{6}+\frac {27}{4} A \,x^{4} b^{6}+\frac {9}{2} x^{6} C \,b^{6}\)	\(550\)
parallelrisch	\(\frac {27}{2} B \,x^{2} a^{3} b^{3}+27 A x \,a^{3} b^{3}+\frac {81}{4} B \,x^{4} a \,b^{5}+\frac {81}{4} C \,x^{4} a^{2} b^{4}+27 A \,a^{2} b^{3} c \,x^{3}+\frac {27}{4} x^{4} A \,a^{2} b^{2} c^{2}+\frac {81}{2} x^{4} A a \,b^{4} c +\frac {81}{4} x^{4} B \,a^{2} b^{3} c +C a b \,c^{4} x^{9}+\frac {27}{5} x^{5} B \,a^{2} b^{2} c^{2}+\frac {162}{5} x^{5} B a \,b^{4} c +\frac {81}{5} x^{5} C \,a^{2} b^{3} c +\frac {9}{2} x^{6} C \,a^{2} b^{2} c^{2}+27 x^{6} C a \,b^{4} c +27 x^{5} A a \,b^{3} c^{2}+9 x^{6} A a \,b^{2} c^{3}+\frac {45}{2} x^{6} B a \,b^{3} c^{2}+\frac {54}{7} x^{7} B a \,b^{2} c^{3}+\frac {135}{7} x^{7} C a \,b^{3} c^{2}+\frac {9}{7} x^{7} A a b \,c^{4}+\frac {9}{8} x^{8} B a b \,c^{4}+\frac {27}{4} x^{8} C a \,b^{2} c^{3}+4 B \,b^{2} c^{4} x^{9}+9 C \,b^{3} c^{3} x^{9}+A b \,c^{5} x^{9}+18 x^{6} A \,b^{4} c^{2}+\frac {27}{2} x^{6} B \,b^{5} c +\frac {81}{5} x^{5} A \,b^{5} c +\frac {108}{7} x^{7} B \,b^{4} c^{2}+\frac {81}{7} x^{7} C \,b^{5} c +\frac {81}{7} x^{7} A \,b^{3} c^{3}+\frac {27}{2} x^{8} C \,b^{4} c^{2}+\frac {9}{11} x^{11} C b \,c^{5}+\frac {9}{10} x^{10} B b \,c^{5}+\frac {18}{5} x^{10} C \,b^{2} c^{4}+\frac {9}{2} x^{8} A \,b^{2} c^{4}+\frac {81}{8} x^{8} B \,b^{3} c^{3}+\frac {81}{5} C \,b^{5} a \,x^{5}+27 A \,x^{3} a \,b^{5}+\frac {81}{2} A \,x^{2} a^{2} b^{4}+27 B \,x^{3} a^{2} b^{4}+9 C \,x^{3} a^{3} b^{3}+\frac {1}{12} C \,c^{6} x^{12}+\frac {1}{11} x^{11} B \,c^{6}+\frac {1}{10} x^{10} A \,c^{6}+\frac {27}{5} x^{5} B \,b^{6}+\frac {27}{4} A \,x^{4} b^{6}+\frac {9}{2} x^{6} C \,b^{6}\)	\(550\)
orering	\(\frac {x \left (770 C \,c^{6} x^{11}+840 B \,c^{6} x^{10}+7560 C b \,c^{5} x^{10}+924 A \,c^{6} x^{9}+8316 B b \,c^{5} x^{9}+33264 C \,b^{2} c^{4} x^{9}+9240 A b \,c^{5} x^{8}+36960 B \,b^{2} c^{4} x^{8}+9240 C a b \,c^{4} x^{8}+83160 C \,b^{3} c^{3} x^{8}+41580 A \,b^{2} c^{4} x^{7}+10395 B a b \,c^{4} x^{7}+93555 B \,b^{3} c^{3} x^{7}+62370 C a \,b^{2} c^{3} x^{7}+124740 C \,b^{4} c^{2} x^{7}+11880 A a b \,c^{4} x^{6}+106920 A \,b^{3} c^{3} x^{6}+71280 B a \,b^{2} c^{3} x^{6}+142560 B \,b^{4} c^{2} x^{6}+178200 C a \,b^{3} c^{2} x^{6}+106920 C \,b^{5} c \,x^{6}+83160 A a \,b^{2} c^{3} x^{5}+166320 A \,b^{4} c^{2} x^{5}+207900 B a \,b^{3} c^{2} x^{5}+124740 B \,b^{5} c \,x^{5}+41580 C \,a^{2} b^{2} c^{2} x^{5}+249480 C a \,b^{4} c \,x^{5}+41580 C \,b^{6} x^{5}+249480 A a \,b^{3} c^{2} x^{4}+149688 A \,b^{5} c \,x^{4}+49896 B \,a^{2} b^{2} c^{2} x^{4}+299376 B a \,b^{4} c \,x^{4}+49896 B \,b^{6} x^{4}+149688 C \,a^{2} b^{3} c \,x^{4}+149688 C a \,b^{5} x^{4}+62370 A \,a^{2} b^{2} c^{2} x^{3}+374220 A a \,b^{4} c \,x^{3}+62370 A \,b^{6} x^{3}+187110 B \,a^{2} b^{3} c \,x^{3}+187110 B a \,b^{5} x^{3}+187110 C \,a^{2} b^{4} x^{3}+249480 A \,a^{2} b^{3} c \,x^{2}+249480 A a \,b^{5} x^{2}+249480 B \,a^{2} b^{4} x^{2}+83160 C \,a^{3} b^{3} x^{2}+374220 A \,a^{2} b^{4} x +124740 B \,a^{3} b^{3} x +249480 A \,a^{3} b^{3}\right )}{9240}\)	\(550\)
default	\(\frac {C \,c^{6} x^{12}}{12}+\frac {\left (B \,c^{6}+9 C b \,c^{5}\right ) x^{11}}{11}+\frac {\left (A \,c^{6}+9 B b \,c^{5}+36 C \,b^{2} c^{4}\right ) x^{10}}{10}+\frac {\left (9 A b \,c^{5}+36 B \,b^{2} c^{4}+C \left (3 a b \,c^{4}+63 b^{3} c^{3}+c^{2} \left (6 a b \,c^{2}+18 b^{3} c \right )\right )\right ) x^{9}}{9}+\frac {\left (36 A \,b^{2} c^{4}+B \left (3 a b \,c^{4}+63 b^{3} c^{3}+c^{2} \left (6 a b \,c^{2}+18 b^{3} c \right )\right )+C \left (18 a \,b^{2} c^{3}+45 b^{4} c^{2}+3 b c \left (6 a b \,c^{2}+18 b^{3} c \right )+c^{2} \left (18 a \,b^{2} c +9 b^{4}\right )\right )\right ) x^{8}}{8}+\frac {\left (A \left (3 a b \,c^{4}+63 b^{3} c^{3}+c^{2} \left (6 a b \,c^{2}+18 b^{3} c \right )\right )+B \left (18 a \,b^{2} c^{3}+45 b^{4} c^{2}+3 b c \left (6 a b \,c^{2}+18 b^{3} c \right )+c^{2} \left (18 a \,b^{2} c +9 b^{4}\right )\right )+C \left (63 a \,b^{3} c^{2}+3 b^{2} \left (6 a b \,c^{2}+18 b^{3} c \right )+3 b c \left (18 a \,b^{2} c +9 b^{4}\right )\right )\right ) x^{7}}{7}+\frac {\left (A \left (18 a \,b^{2} c^{3}+45 b^{4} c^{2}+3 b c \left (6 a b \,c^{2}+18 b^{3} c \right )+c^{2} \left (18 a \,b^{2} c +9 b^{4}\right )\right )+B \left (63 a \,b^{3} c^{2}+3 b^{2} \left (6 a b \,c^{2}+18 b^{3} c \right )+3 b c \left (18 a \,b^{2} c +9 b^{4}\right )\right )+C \left (3 a b \left (6 a b \,c^{2}+18 b^{3} c \right )+3 b^{2} \left (18 a \,b^{2} c +9 b^{4}\right )+54 a \,b^{4} c +9 a^{2} b^{2} c^{2}\right )\right ) x^{6}}{6}+\frac {\left (A \left (63 a \,b^{3} c^{2}+3 b^{2} \left (6 a b \,c^{2}+18 b^{3} c \right )+3 b c \left (18 a \,b^{2} c +9 b^{4}\right )\right )+B \left (3 a b \left (6 a b \,c^{2}+18 b^{3} c \right )+3 b^{2} \left (18 a \,b^{2} c +9 b^{4}\right )+54 a \,b^{4} c +9 a^{2} b^{2} c^{2}\right )+C \left (3 a b \left (18 a \,b^{2} c +9 b^{4}\right )+54 b^{5} a +27 a^{2} b^{3} c \right )\right ) x^{5}}{5}+\frac {\left (A \left (3 a b \left (6 a b \,c^{2}+18 b^{3} c \right )+3 b^{2} \left (18 a \,b^{2} c +9 b^{4}\right )+54 a \,b^{4} c +9 a^{2} b^{2} c^{2}\right )+B \left (3 a b \left (18 a \,b^{2} c +9 b^{4}\right )+54 b^{5} a +27 a^{2} b^{3} c \right )+81 b^{4} C \,a^{2}\right ) x^{4}}{4}+\frac {\left (A \left (3 a b \left (18 a \,b^{2} c +9 b^{4}\right )+54 b^{5} a +27 a^{2} b^{3} c \right )+81 B \,b^{4} a^{2}+27 C \,b^{3} a^{3}\right ) x^{3}}{3}+\frac {\left (81 A \,a^{2} b^{4}+27 B \,a^{3} b^{3}\right ) x^{2}}{2}+27 A x \,a^{3} b^{3}\)	\(877\)

Input:

int((C*x^2+B*x+A)*(c^2*x^3+3*b*c*x^2+3*b^2*x+3*a*b)^3,x,method=_RETURNVERB 
OSE)
 

Output:

1/12*C*c^6*x^12+(1/11*B*c^6+9/11*C*b*c^5)*x^11+(1/10*A*c^6+9/10*B*b*c^5+18 
/5*C*b^2*c^4)*x^10+(A*b*c^5+4*B*b^2*c^4+C*a*b*c^4+9*C*b^3*c^3)*x^9+(9/2*A* 
b^2*c^4+9/8*B*a*b*c^4+81/8*B*b^3*c^3+27/4*C*a*b^2*c^3+27/2*C*b^4*c^2)*x^8+ 
(9/7*A*a*b*c^4+81/7*A*b^3*c^3+54/7*B*a*b^2*c^3+108/7*B*b^4*c^2+135/7*C*a*b 
^3*c^2+81/7*C*b^5*c)*x^7+(9*A*a*b^2*c^3+18*A*b^4*c^2+45/2*B*a*b^3*c^2+27/2 
*B*b^5*c+9/2*C*a^2*b^2*c^2+27*C*a*b^4*c+9/2*C*b^6)*x^6+(27*A*a*b^3*c^2+81/ 
5*A*b^5*c+27/5*B*a^2*b^2*c^2+162/5*B*a*b^4*c+27/5*B*b^6+81/5*C*a^2*b^3*c+8 
1/5*C*b^5*a)*x^5+(27/4*A*a^2*b^2*c^2+81/2*A*a*b^4*c+27/4*A*b^6+81/4*B*a^2* 
b^3*c+81/4*B*b^5*a+81/4*b^4*C*a^2)*x^4+(27*A*a^2*b^3*c+27*A*a*b^5+27*B*a^2 
*b^4+9*C*a^3*b^3)*x^3+(81/2*A*a^2*b^4+27/2*B*a^3*b^3)*x^2+27*A*x*a^3*b^3
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.46 \[ \int \left (A+B x+C x^2\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=\frac {1}{12} \, C c^{6} x^{12} + \frac {1}{11} \, {\left (9 \, C b c^{5} + B c^{6}\right )} x^{11} + \frac {1}{10} \, {\left (36 \, C b^{2} c^{4} + 9 \, B b c^{5} + A c^{6}\right )} x^{10} + {\left (9 \, C b^{3} c^{3} + A b c^{5} + {\left (C a b + 4 \, B b^{2}\right )} c^{4}\right )} x^{9} + \frac {9}{8} \, {\left (12 \, C b^{4} c^{2} + {\left (B a b + 4 \, A b^{2}\right )} c^{4} + 3 \, {\left (2 \, C a b^{2} + 3 \, B b^{3}\right )} c^{3}\right )} x^{8} + 27 \, A a^{3} b^{3} x + \frac {9}{7} \, {\left (9 \, C b^{5} c + A a b c^{4} + 3 \, {\left (2 \, B a b^{2} + 3 \, A b^{3}\right )} c^{3} + 3 \, {\left (5 \, C a b^{3} + 4 \, B b^{4}\right )} c^{2}\right )} x^{7} + \frac {9}{2} \, {\left (C b^{6} + 2 \, A a b^{2} c^{3} + {\left (C a^{2} b^{2} + 5 \, B a b^{3} + 4 \, A b^{4}\right )} c^{2} + 3 \, {\left (2 \, C a b^{4} + B b^{5}\right )} c\right )} x^{6} + \frac {27}{5} \, {\left (3 \, C a b^{5} + B b^{6} + {\left (B a^{2} b^{2} + 5 \, A a b^{3}\right )} c^{2} + 3 \, {\left (C a^{2} b^{3} + 2 \, B a b^{4} + A b^{5}\right )} c\right )} x^{5} + \frac {27}{4} \, {\left (3 \, C a^{2} b^{4} + 3 \, B a b^{5} + A b^{6} + A a^{2} b^{2} c^{2} + 3 \, {\left (B a^{2} b^{3} + 2 \, A a b^{4}\right )} c\right )} x^{4} + 9 \, {\left (C a^{3} b^{3} + 3 \, B a^{2} b^{4} + 3 \, A a b^{5} + 3 \, A a^{2} b^{3} c\right )} x^{3} + \frac {27}{2} \, {\left (B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{2} \] Input:

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

Output:

1/12*C*c^6*x^12 + 1/11*(9*C*b*c^5 + B*c^6)*x^11 + 1/10*(36*C*b^2*c^4 + 9*B 
*b*c^5 + A*c^6)*x^10 + (9*C*b^3*c^3 + A*b*c^5 + (C*a*b + 4*B*b^2)*c^4)*x^9 
 + 9/8*(12*C*b^4*c^2 + (B*a*b + 4*A*b^2)*c^4 + 3*(2*C*a*b^2 + 3*B*b^3)*c^3 
)*x^8 + 27*A*a^3*b^3*x + 9/7*(9*C*b^5*c + A*a*b*c^4 + 3*(2*B*a*b^2 + 3*A*b 
^3)*c^3 + 3*(5*C*a*b^3 + 4*B*b^4)*c^2)*x^7 + 9/2*(C*b^6 + 2*A*a*b^2*c^3 + 
(C*a^2*b^2 + 5*B*a*b^3 + 4*A*b^4)*c^2 + 3*(2*C*a*b^4 + B*b^5)*c)*x^6 + 27/ 
5*(3*C*a*b^5 + B*b^6 + (B*a^2*b^2 + 5*A*a*b^3)*c^2 + 3*(C*a^2*b^3 + 2*B*a* 
b^4 + A*b^5)*c)*x^5 + 27/4*(3*C*a^2*b^4 + 3*B*a*b^5 + A*b^6 + A*a^2*b^2*c^ 
2 + 3*(B*a^2*b^3 + 2*A*a*b^4)*c)*x^4 + 9*(C*a^3*b^3 + 3*B*a^2*b^4 + 3*A*a* 
b^5 + 3*A*a^2*b^3*c)*x^3 + 27/2*(B*a^3*b^3 + 3*A*a^2*b^4)*x^2
 

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.83 \[ \int \left (A+B x+C x^2\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=27 A a^{3} b^{3} x + \frac {C c^{6} x^{12}}{12} + x^{11} \left (\frac {B c^{6}}{11} + \frac {9 C b c^{5}}{11}\right ) + x^{10} \left (\frac {A c^{6}}{10} + \frac {9 B b c^{5}}{10} + \frac {18 C b^{2} c^{4}}{5}\right ) + x^{9} \left (A b c^{5} + 4 B b^{2} c^{4} + C a b c^{4} + 9 C b^{3} c^{3}\right ) + x^{8} \cdot \left (\frac {9 A b^{2} c^{4}}{2} + \frac {9 B a b c^{4}}{8} + \frac {81 B b^{3} c^{3}}{8} + \frac {27 C a b^{2} c^{3}}{4} + \frac {27 C b^{4} c^{2}}{2}\right ) + x^{7} \cdot \left (\frac {9 A a b c^{4}}{7} + \frac {81 A b^{3} c^{3}}{7} + \frac {54 B a b^{2} c^{3}}{7} + \frac {108 B b^{4} c^{2}}{7} + \frac {135 C a b^{3} c^{2}}{7} + \frac {81 C b^{5} c}{7}\right ) + x^{6} \cdot \left (9 A a b^{2} c^{3} + 18 A b^{4} c^{2} + \frac {45 B a b^{3} c^{2}}{2} + \frac {27 B b^{5} c}{2} + \frac {9 C a^{2} b^{2} c^{2}}{2} + 27 C a b^{4} c + \frac {9 C b^{6}}{2}\right ) + x^{5} \cdot \left (27 A a b^{3} c^{2} + \frac {81 A b^{5} c}{5} + \frac {27 B a^{2} b^{2} c^{2}}{5} + \frac {162 B a b^{4} c}{5} + \frac {27 B b^{6}}{5} + \frac {81 C a^{2} b^{3} c}{5} + \frac {81 C a b^{5}}{5}\right ) + x^{4} \cdot \left (\frac {27 A a^{2} b^{2} c^{2}}{4} + \frac {81 A a b^{4} c}{2} + \frac {27 A b^{6}}{4} + \frac {81 B a^{2} b^{3} c}{4} + \frac {81 B a b^{5}}{4} + \frac {81 C a^{2} b^{4}}{4}\right ) + x^{3} \cdot \left (27 A a^{2} b^{3} c + 27 A a b^{5} + 27 B a^{2} b^{4} + 9 C a^{3} b^{3}\right ) + x^{2} \cdot \left (\frac {81 A a^{2} b^{4}}{2} + \frac {27 B a^{3} b^{3}}{2}\right ) \] Input:

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

Output:

27*A*a**3*b**3*x + C*c**6*x**12/12 + x**11*(B*c**6/11 + 9*C*b*c**5/11) + x 
**10*(A*c**6/10 + 9*B*b*c**5/10 + 18*C*b**2*c**4/5) + x**9*(A*b*c**5 + 4*B 
*b**2*c**4 + C*a*b*c**4 + 9*C*b**3*c**3) + x**8*(9*A*b**2*c**4/2 + 9*B*a*b 
*c**4/8 + 81*B*b**3*c**3/8 + 27*C*a*b**2*c**3/4 + 27*C*b**4*c**2/2) + x**7 
*(9*A*a*b*c**4/7 + 81*A*b**3*c**3/7 + 54*B*a*b**2*c**3/7 + 108*B*b**4*c**2 
/7 + 135*C*a*b**3*c**2/7 + 81*C*b**5*c/7) + x**6*(9*A*a*b**2*c**3 + 18*A*b 
**4*c**2 + 45*B*a*b**3*c**2/2 + 27*B*b**5*c/2 + 9*C*a**2*b**2*c**2/2 + 27* 
C*a*b**4*c + 9*C*b**6/2) + x**5*(27*A*a*b**3*c**2 + 81*A*b**5*c/5 + 27*B*a 
**2*b**2*c**2/5 + 162*B*a*b**4*c/5 + 27*B*b**6/5 + 81*C*a**2*b**3*c/5 + 81 
*C*a*b**5/5) + x**4*(27*A*a**2*b**2*c**2/4 + 81*A*a*b**4*c/2 + 27*A*b**6/4 
 + 81*B*a**2*b**3*c/4 + 81*B*a*b**5/4 + 81*C*a**2*b**4/4) + x**3*(27*A*a** 
2*b**3*c + 27*A*a*b**5 + 27*B*a**2*b**4 + 9*C*a**3*b**3) + x**2*(81*A*a**2 
*b**4/2 + 27*B*a**3*b**3/2)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.46 \[ \int \left (A+B x+C x^2\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=\frac {1}{12} \, C c^{6} x^{12} + \frac {1}{11} \, {\left (9 \, C b c^{5} + B c^{6}\right )} x^{11} + \frac {1}{10} \, {\left (36 \, C b^{2} c^{4} + 9 \, B b c^{5} + A c^{6}\right )} x^{10} + {\left (9 \, C b^{3} c^{3} + A b c^{5} + {\left (C a b + 4 \, B b^{2}\right )} c^{4}\right )} x^{9} + \frac {9}{8} \, {\left (12 \, C b^{4} c^{2} + {\left (B a b + 4 \, A b^{2}\right )} c^{4} + 3 \, {\left (2 \, C a b^{2} + 3 \, B b^{3}\right )} c^{3}\right )} x^{8} + 27 \, A a^{3} b^{3} x + \frac {9}{7} \, {\left (9 \, C b^{5} c + A a b c^{4} + 3 \, {\left (2 \, B a b^{2} + 3 \, A b^{3}\right )} c^{3} + 3 \, {\left (5 \, C a b^{3} + 4 \, B b^{4}\right )} c^{2}\right )} x^{7} + \frac {9}{2} \, {\left (C b^{6} + 2 \, A a b^{2} c^{3} + {\left (C a^{2} b^{2} + 5 \, B a b^{3} + 4 \, A b^{4}\right )} c^{2} + 3 \, {\left (2 \, C a b^{4} + B b^{5}\right )} c\right )} x^{6} + \frac {27}{5} \, {\left (3 \, C a b^{5} + B b^{6} + {\left (B a^{2} b^{2} + 5 \, A a b^{3}\right )} c^{2} + 3 \, {\left (C a^{2} b^{3} + 2 \, B a b^{4} + A b^{5}\right )} c\right )} x^{5} + \frac {27}{4} \, {\left (3 \, C a^{2} b^{4} + 3 \, B a b^{5} + A b^{6} + A a^{2} b^{2} c^{2} + 3 \, {\left (B a^{2} b^{3} + 2 \, A a b^{4}\right )} c\right )} x^{4} + 9 \, {\left (C a^{3} b^{3} + 3 \, B a^{2} b^{4} + 3 \, A a b^{5} + 3 \, A a^{2} b^{3} c\right )} x^{3} + \frac {27}{2} \, {\left (B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{2} \] Input:

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

Output:

1/12*C*c^6*x^12 + 1/11*(9*C*b*c^5 + B*c^6)*x^11 + 1/10*(36*C*b^2*c^4 + 9*B 
*b*c^5 + A*c^6)*x^10 + (9*C*b^3*c^3 + A*b*c^5 + (C*a*b + 4*B*b^2)*c^4)*x^9 
 + 9/8*(12*C*b^4*c^2 + (B*a*b + 4*A*b^2)*c^4 + 3*(2*C*a*b^2 + 3*B*b^3)*c^3 
)*x^8 + 27*A*a^3*b^3*x + 9/7*(9*C*b^5*c + A*a*b*c^4 + 3*(2*B*a*b^2 + 3*A*b 
^3)*c^3 + 3*(5*C*a*b^3 + 4*B*b^4)*c^2)*x^7 + 9/2*(C*b^6 + 2*A*a*b^2*c^3 + 
(C*a^2*b^2 + 5*B*a*b^3 + 4*A*b^4)*c^2 + 3*(2*C*a*b^4 + B*b^5)*c)*x^6 + 27/ 
5*(3*C*a*b^5 + B*b^6 + (B*a^2*b^2 + 5*A*a*b^3)*c^2 + 3*(C*a^2*b^3 + 2*B*a* 
b^4 + A*b^5)*c)*x^5 + 27/4*(3*C*a^2*b^4 + 3*B*a*b^5 + A*b^6 + A*a^2*b^2*c^ 
2 + 3*(B*a^2*b^3 + 2*A*a*b^4)*c)*x^4 + 9*(C*a^3*b^3 + 3*B*a^2*b^4 + 3*A*a* 
b^5 + 3*A*a^2*b^3*c)*x^3 + 27/2*(B*a^3*b^3 + 3*A*a^2*b^4)*x^2
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.77 \[ \int \left (A+B x+C x^2\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=\frac {1}{12} \, C c^{6} x^{12} + \frac {9}{11} \, C b c^{5} x^{11} + \frac {1}{11} \, B c^{6} x^{11} + \frac {18}{5} \, C b^{2} c^{4} x^{10} + \frac {9}{10} \, B b c^{5} x^{10} + \frac {1}{10} \, A c^{6} x^{10} + 9 \, C b^{3} c^{3} x^{9} + C a b c^{4} x^{9} + 4 \, B b^{2} c^{4} x^{9} + A b c^{5} x^{9} + \frac {27}{2} \, C b^{4} c^{2} x^{8} + \frac {27}{4} \, C a b^{2} c^{3} x^{8} + \frac {81}{8} \, B b^{3} c^{3} x^{8} + \frac {9}{8} \, B a b c^{4} x^{8} + \frac {9}{2} \, A b^{2} c^{4} x^{8} + \frac {81}{7} \, C b^{5} c x^{7} + \frac {135}{7} \, C a b^{3} c^{2} x^{7} + \frac {108}{7} \, B b^{4} c^{2} x^{7} + \frac {54}{7} \, B a b^{2} c^{3} x^{7} + \frac {81}{7} \, A b^{3} c^{3} x^{7} + \frac {9}{7} \, A a b c^{4} x^{7} + \frac {9}{2} \, C b^{6} x^{6} + 27 \, C a b^{4} c x^{6} + \frac {27}{2} \, B b^{5} c x^{6} + \frac {9}{2} \, C a^{2} b^{2} c^{2} x^{6} + \frac {45}{2} \, B a b^{3} c^{2} x^{6} + 18 \, A b^{4} c^{2} x^{6} + 9 \, A a b^{2} c^{3} x^{6} + \frac {81}{5} \, C a b^{5} x^{5} + \frac {27}{5} \, B b^{6} x^{5} + \frac {81}{5} \, C a^{2} b^{3} c x^{5} + \frac {162}{5} \, B a b^{4} c x^{5} + \frac {81}{5} \, A b^{5} c x^{5} + \frac {27}{5} \, B a^{2} b^{2} c^{2} x^{5} + 27 \, A a b^{3} c^{2} x^{5} + \frac {81}{4} \, C a^{2} b^{4} x^{4} + \frac {81}{4} \, B a b^{5} x^{4} + \frac {27}{4} \, A b^{6} x^{4} + \frac {81}{4} \, B a^{2} b^{3} c x^{4} + \frac {81}{2} \, A a b^{4} c x^{4} + \frac {27}{4} \, A a^{2} b^{2} c^{2} x^{4} + 9 \, C a^{3} b^{3} x^{3} + 27 \, B a^{2} b^{4} x^{3} + 27 \, A a b^{5} x^{3} + 27 \, A a^{2} b^{3} c x^{3} + \frac {27}{2} \, B a^{3} b^{3} x^{2} + \frac {81}{2} \, A a^{2} b^{4} x^{2} + 27 \, A a^{3} b^{3} x \] Input:

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

Output:

1/12*C*c^6*x^12 + 9/11*C*b*c^5*x^11 + 1/11*B*c^6*x^11 + 18/5*C*b^2*c^4*x^1 
0 + 9/10*B*b*c^5*x^10 + 1/10*A*c^6*x^10 + 9*C*b^3*c^3*x^9 + C*a*b*c^4*x^9 
+ 4*B*b^2*c^4*x^9 + A*b*c^5*x^9 + 27/2*C*b^4*c^2*x^8 + 27/4*C*a*b^2*c^3*x^ 
8 + 81/8*B*b^3*c^3*x^8 + 9/8*B*a*b*c^4*x^8 + 9/2*A*b^2*c^4*x^8 + 81/7*C*b^ 
5*c*x^7 + 135/7*C*a*b^3*c^2*x^7 + 108/7*B*b^4*c^2*x^7 + 54/7*B*a*b^2*c^3*x 
^7 + 81/7*A*b^3*c^3*x^7 + 9/7*A*a*b*c^4*x^7 + 9/2*C*b^6*x^6 + 27*C*a*b^4*c 
*x^6 + 27/2*B*b^5*c*x^6 + 9/2*C*a^2*b^2*c^2*x^6 + 45/2*B*a*b^3*c^2*x^6 + 1 
8*A*b^4*c^2*x^6 + 9*A*a*b^2*c^3*x^6 + 81/5*C*a*b^5*x^5 + 27/5*B*b^6*x^5 + 
81/5*C*a^2*b^3*c*x^5 + 162/5*B*a*b^4*c*x^5 + 81/5*A*b^5*c*x^5 + 27/5*B*a^2 
*b^2*c^2*x^5 + 27*A*a*b^3*c^2*x^5 + 81/4*C*a^2*b^4*x^4 + 81/4*B*a*b^5*x^4 
+ 27/4*A*b^6*x^4 + 81/4*B*a^2*b^3*c*x^4 + 81/2*A*a*b^4*c*x^4 + 27/4*A*a^2* 
b^2*c^2*x^4 + 9*C*a^3*b^3*x^3 + 27*B*a^2*b^4*x^3 + 27*A*a*b^5*x^3 + 27*A*a 
^2*b^3*c*x^3 + 27/2*B*a^3*b^3*x^2 + 81/2*A*a^2*b^4*x^2 + 27*A*a^3*b^3*x
 

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.40 \[ \int \left (A+B x+C x^2\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=x^4\,\left (\frac {81\,C\,a^2\,b^4}{4}+\frac {81\,B\,a^2\,b^3\,c}{4}+\frac {27\,A\,a^2\,b^2\,c^2}{4}+\frac {81\,B\,a\,b^5}{4}+\frac {81\,A\,a\,b^4\,c}{2}+\frac {27\,A\,b^6}{4}\right )+x^5\,\left (\frac {81\,C\,a^2\,b^3\,c}{5}+\frac {27\,B\,a^2\,b^2\,c^2}{5}+\frac {81\,C\,a\,b^5}{5}+\frac {162\,B\,a\,b^4\,c}{5}+27\,A\,a\,b^3\,c^2+\frac {27\,B\,b^6}{5}+\frac {81\,A\,b^5\,c}{5}\right )+x^{11}\,\left (\frac {B\,c^6}{11}+\frac {9\,C\,b\,c^5}{11}\right )+x^7\,\left (\frac {81\,C\,b^5\,c}{7}+\frac {108\,B\,b^4\,c^2}{7}+\frac {81\,A\,b^3\,c^3}{7}+\frac {135\,C\,a\,b^3\,c^2}{7}+\frac {54\,B\,a\,b^2\,c^3}{7}+\frac {9\,A\,a\,b\,c^4}{7}\right )+x^{10}\,\left (\frac {18\,C\,b^2\,c^4}{5}+\frac {9\,B\,b\,c^5}{10}+\frac {A\,c^6}{10}\right )+x^6\,\left (\frac {9\,C\,a^2\,b^2\,c^2}{2}+27\,C\,a\,b^4\,c+\frac {45\,B\,a\,b^3\,c^2}{2}+9\,A\,a\,b^2\,c^3+\frac {9\,C\,b^6}{2}+\frac {27\,B\,b^5\,c}{2}+18\,A\,b^4\,c^2\right )+\frac {C\,c^6\,x^{12}}{12}+\frac {27\,a^2\,b^3\,x^2\,\left (3\,A\,b+B\,a\right )}{2}+\frac {9\,b\,c^2\,x^8\,\left (12\,C\,b^3+9\,B\,b^2\,c+4\,A\,b\,c^2+6\,C\,a\,b\,c+B\,a\,c^2\right )}{8}+9\,a\,b^3\,x^3\,\left (C\,a^2+3\,B\,a\,b+3\,A\,c\,a+3\,A\,b^2\right )+b\,c^3\,x^9\,\left (9\,C\,b^2+4\,B\,b\,c+A\,c^2+C\,a\,c\right )+27\,A\,a^3\,b^3\,x \] Input:

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

Output:

x^4*((27*A*b^6)/4 + (81*C*a^2*b^4)/4 + (81*B*a*b^5)/4 + (81*A*a*b^4*c)/2 + 
 (81*B*a^2*b^3*c)/4 + (27*A*a^2*b^2*c^2)/4) + x^5*((27*B*b^6)/5 + (81*A*b^ 
5*c)/5 + (81*C*a*b^5)/5 + (162*B*a*b^4*c)/5 + 27*A*a*b^3*c^2 + (81*C*a^2*b 
^3*c)/5 + (27*B*a^2*b^2*c^2)/5) + x^11*((B*c^6)/11 + (9*C*b*c^5)/11) + x^7 
*((81*A*b^3*c^3)/7 + (108*B*b^4*c^2)/7 + (81*C*b^5*c)/7 + (9*A*a*b*c^4)/7 
+ (54*B*a*b^2*c^3)/7 + (135*C*a*b^3*c^2)/7) + x^10*((A*c^6)/10 + (18*C*b^2 
*c^4)/5 + (9*B*b*c^5)/10) + x^6*((9*C*b^6)/2 + 18*A*b^4*c^2 + (27*B*b^5*c) 
/2 + 27*C*a*b^4*c + 9*A*a*b^2*c^3 + (45*B*a*b^3*c^2)/2 + (9*C*a^2*b^2*c^2) 
/2) + (C*c^6*x^12)/12 + (27*a^2*b^3*x^2*(3*A*b + B*a))/2 + (9*b*c^2*x^8*(1 
2*C*b^3 + 4*A*b*c^2 + B*a*c^2 + 9*B*b^2*c + 6*C*a*b*c))/8 + 9*a*b^3*x^3*(3 
*A*b^2 + C*a^2 + 3*A*a*c + 3*B*a*b) + b*c^3*x^9*(A*c^2 + 9*C*b^2 + 4*B*b*c 
 + C*a*c) + 27*A*a^3*b^3*x
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.85 \[ \int \left (A+B x+C x^2\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=\frac {x \left (770 c^{7} x^{11}+8400 b \,c^{6} x^{10}+924 a \,c^{6} x^{9}+41580 b^{2} c^{5} x^{9}+18480 a b \,c^{5} x^{8}+120120 b^{3} c^{4} x^{8}+114345 a \,b^{2} c^{4} x^{7}+218295 b^{4} c^{3} x^{7}+11880 a^{2} b \,c^{4} x^{6}+356400 a \,b^{3} c^{3} x^{6}+249480 b^{5} c^{2} x^{6}+124740 a^{2} b^{2} c^{3} x^{5}+623700 a \,b^{4} c^{2} x^{5}+166320 b^{6} c \,x^{5}+449064 a^{2} b^{3} c^{2} x^{4}+598752 a \,b^{5} c \,x^{4}+49896 b^{7} x^{4}+62370 a^{3} b^{2} c^{2} x^{3}+748440 a^{2} b^{4} c \,x^{3}+249480 a \,b^{6} x^{3}+332640 a^{3} b^{3} c \,x^{2}+498960 a^{2} b^{5} x^{2}+498960 a^{3} b^{4} x +249480 a^{4} b^{3}\right )}{9240} \] Input:

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

Output:

(x*(249480*a**4*b**3 + 498960*a**3*b**4*x + 332640*a**3*b**3*c*x**2 + 6237 
0*a**3*b**2*c**2*x**3 + 498960*a**2*b**5*x**2 + 748440*a**2*b**4*c*x**3 + 
449064*a**2*b**3*c**2*x**4 + 124740*a**2*b**2*c**3*x**5 + 11880*a**2*b*c** 
4*x**6 + 249480*a*b**6*x**3 + 598752*a*b**5*c*x**4 + 623700*a*b**4*c**2*x* 
*5 + 356400*a*b**3*c**3*x**6 + 114345*a*b**2*c**4*x**7 + 18480*a*b*c**5*x* 
*8 + 924*a*c**6*x**9 + 49896*b**7*x**4 + 166320*b**6*c*x**5 + 249480*b**5* 
c**2*x**6 + 218295*b**4*c**3*x**7 + 120120*b**3*c**4*x**8 + 41580*b**2*c** 
5*x**9 + 8400*b*c**6*x**10 + 770*c**7*x**11))/9240