\(\int (A+B x+C x^2) (2-6 x^2+3 x^3)^3 \, dx\) [49]

Optimal result

Integrand size = 25, antiderivative size = 145 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^3 \, dx=8 A x+4 B x^2-\frac {8}{3} (9 A-C) x^3+9 (A-2 B) x^4+\frac {36}{5} (6 A+B-2 C) x^5-6 (6 A-6 B-C) x^6-\frac {54}{7} (3 A+4 B-4 C) x^7+\frac {27}{4} (6 A-3 B-4 C) x^8-18 (A-2 B+C) x^9+\frac {27}{10} (A-6 B+12 C) x^{10}+\frac {27}{11} (B-6 C) x^{11}+\frac {9 C x^{12}}{4} \] Output:

8*A*x+4*B*x^2-8/3*(9*A-C)*x^3+9*(A-2*B)*x^4+36/5*(6*A+B-2*C)*x^5-6*(6*A-6* 
B-C)*x^6-54/7*(3*A+4*B-4*C)*x^7+27/4*(6*A-3*B-4*C)*x^8-18*(A-2*B+C)*x^9+27 
/10*(A-6*B+12*C)*x^10+27/11*(B-6*C)*x^11+9/4*C*x^12
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^3 \, dx=8 A x+4 B x^2-\frac {8}{3} (9 A-C) x^3+9 (A-2 B) x^4+\frac {36}{5} (6 A+B-2 C) x^5-6 (6 A-6 B-C) x^6-\frac {54}{7} (3 A+4 B-4 C) x^7+\frac {27}{4} (6 A-3 B-4 C) x^8-18 (A-2 B+C) x^9+\frac {27}{10} (A-6 B+12 C) x^{10}+\frac {27}{11} (B-6 C) x^{11}+\frac {9 C x^{12}}{4} \] Input:

Integrate[(A + B*x + C*x^2)*(2 - 6*x^2 + 3*x^3)^3,x]
 

Output:

8*A*x + 4*B*x^2 - (8*(9*A - C)*x^3)/3 + 9*(A - 2*B)*x^4 + (36*(6*A + B - 2 
*C)*x^5)/5 - 6*(6*A - 6*B - C)*x^6 - (54*(3*A + 4*B - 4*C)*x^7)/7 + (27*(6 
*A - 3*B - 4*C)*x^8)/4 - 18*(A - 2*B + C)*x^9 + (27*(A - 6*B + 12*C)*x^10) 
/10 + (27*(B - 6*C)*x^11)/11 + (9*C*x^12)/4
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.

\(\displaystyle \int \left (3 x^3-6 x^2+2\right )^3 \left (A+B x+C x^2\right ) \, dx\)

\(\Big \downarrow \) 2188

\(\displaystyle \int \left (27 x^9 (A-6 B+12 C)-162 x^8 (A-2 B+C)+54 x^7 (6 A-3 B-4 C)-54 x^6 (3 A+4 B-4 C)-36 x^5 (6 A-6 B-C)+36 x^4 (6 A+B-2 C)+36 x^3 (A-2 B)-8 x^2 (9 A-C)+8 A+27 x^{10} (B-6 C)+8 B x+27 C x^{11}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {27}{10} x^{10} (A-6 B+12 C)-18 x^9 (A-2 B+C)+\frac {27}{4} x^8 (6 A-3 B-4 C)-\frac {54}{7} x^7 (3 A+4 B-4 C)-6 x^6 (6 A-6 B-C)+\frac {36}{5} x^5 (6 A+B-2 C)+9 x^4 (A-2 B)-\frac {8}{3} x^3 (9 A-C)+8 A x+\frac {27}{11} x^{11} (B-6 C)+4 B x^2+\frac {9 C x^{12}}{4}\)

Input:

Int[(A + B*x + C*x^2)*(2 - 6*x^2 + 3*x^3)^3,x]
 

Output:

8*A*x + 4*B*x^2 - (8*(9*A - C)*x^3)/3 + 9*(A - 2*B)*x^4 + (36*(6*A + B - 2 
*C)*x^5)/5 - 6*(6*A - 6*B - C)*x^6 - (54*(3*A + 4*B - 4*C)*x^7)/7 + (27*(6 
*A - 3*B - 4*C)*x^8)/4 - 18*(A - 2*B + C)*x^9 + (27*(A - 6*B + 12*C)*x^10) 
/10 + (27*(B - 6*C)*x^11)/11 + (9*C*x^12)/4
 

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 = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.93

method	result	size
norman	\(\frac {9 C \,x^{12}}{4}+\left (\frac {27 B}{11}-\frac {162 C}{11}\right ) x^{11}+\left (\frac {27 A}{10}-\frac {81 B}{5}+\frac {162 C}{5}\right ) x^{10}+\left (-18 A +36 B -18 C \right ) x^{9}+\left (\frac {81 A}{2}-\frac {81 B}{4}-27 C \right ) x^{8}+\left (-\frac {162 A}{7}-\frac {216 B}{7}+\frac {216 C}{7}\right ) x^{7}+\left (-36 A +36 B +6 C \right ) x^{6}+\left (\frac {216 A}{5}+\frac {36 B}{5}-\frac {72 C}{5}\right ) x^{5}+\left (9 A -18 B \right ) x^{4}+\left (-24 A +\frac {8 C}{3}\right ) x^{3}+4 B \,x^{2}+8 A x\)	\(135\)
default	\(\frac {9 C \,x^{12}}{4}+\frac {\left (27 B -162 C \right ) x^{11}}{11}+\frac {\left (27 A -162 B +324 C \right ) x^{10}}{10}+\frac {\left (-162 A +324 B -162 C \right ) x^{9}}{9}+\frac {\left (324 A -162 B -216 C \right ) x^{8}}{8}+\frac {\left (-162 A -216 B +216 C \right ) x^{7}}{7}+\frac {\left (-216 A +216 B +36 C \right ) x^{6}}{6}+\frac {\left (216 A +36 B -72 C \right ) x^{5}}{5}+\frac {\left (36 A -72 B \right ) x^{4}}{4}+\frac {\left (-72 A +8 C \right ) x^{3}}{3}+4 B \,x^{2}+8 A x\)	\(144\)
gosper	\(\frac {8}{3} C \,x^{3}+4 B \,x^{2}+9 x^{4} A +\frac {216}{5} x^{5} A -18 A \,x^{9}-\frac {81}{5} B \,x^{10}-27 x^{8} C -\frac {162}{7} x^{7} A +36 x^{9} B +\frac {81}{2} x^{8} A -\frac {216}{7} x^{7} B +\frac {216}{7} x^{7} C -36 x^{6} A +36 x^{6} B +\frac {27}{10} x^{10} A +\frac {162}{5} x^{10} C +8 A x +\frac {27}{11} B \,x^{11}+\frac {9}{4} C \,x^{12}-\frac {81}{4} B \,x^{8}+\frac {36}{5} B \,x^{5}+6 C \,x^{6}-\frac {162}{11} C \,x^{11}-18 C \,x^{9}-24 x^{3} A -18 x^{4} B -\frac {72}{5} x^{5} C\)	\(162\)
risch	\(\frac {8}{3} C \,x^{3}+4 B \,x^{2}+9 x^{4} A +\frac {216}{5} x^{5} A -18 A \,x^{9}-\frac {81}{5} B \,x^{10}-27 x^{8} C -\frac {162}{7} x^{7} A +36 x^{9} B +\frac {81}{2} x^{8} A -\frac {216}{7} x^{7} B +\frac {216}{7} x^{7} C -36 x^{6} A +36 x^{6} B +\frac {27}{10} x^{10} A +\frac {162}{5} x^{10} C +8 A x +\frac {27}{11} B \,x^{11}+\frac {9}{4} C \,x^{12}-\frac {81}{4} B \,x^{8}+\frac {36}{5} B \,x^{5}+6 C \,x^{6}-\frac {162}{11} C \,x^{11}-18 C \,x^{9}-24 x^{3} A -18 x^{4} B -\frac {72}{5} x^{5} C\)	\(162\)
parallelrisch	\(\frac {8}{3} C \,x^{3}+4 B \,x^{2}+9 x^{4} A +\frac {216}{5} x^{5} A -18 A \,x^{9}-\frac {81}{5} B \,x^{10}-27 x^{8} C -\frac {162}{7} x^{7} A +36 x^{9} B +\frac {81}{2} x^{8} A -\frac {216}{7} x^{7} B +\frac {216}{7} x^{7} C -36 x^{6} A +36 x^{6} B +\frac {27}{10} x^{10} A +\frac {162}{5} x^{10} C +8 A x +\frac {27}{11} B \,x^{11}+\frac {9}{4} C \,x^{12}-\frac {81}{4} B \,x^{8}+\frac {36}{5} B \,x^{5}+6 C \,x^{6}-\frac {162}{11} C \,x^{11}-18 C \,x^{9}-24 x^{3} A -18 x^{4} B -\frac {72}{5} x^{5} C\)	\(162\)
orering	\(\frac {x \left (10395 C \,x^{11}+11340 B \,x^{10}-68040 x^{10} C +12474 A \,x^{9}-74844 x^{9} B +149688 C \,x^{9}-83160 x^{8} A +166320 B \,x^{8}-83160 x^{8} C +187110 x^{7} A -93555 x^{7} B -124740 x^{7} C -106920 x^{6} A -142560 x^{6} B +142560 C \,x^{6}-166320 x^{5} A +166320 B \,x^{5}+27720 x^{5} C +199584 x^{4} A +33264 x^{4} B -66528 C \,x^{4}+41580 x^{3} A -83160 B \,x^{3}-110880 A \,x^{2}+12320 C \,x^{2}+18480 B x +36960 A \right )}{4620}\)	\(162\)

Input:

int((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^3,x,method=_RETURNVERBOSE)
 

Output:

9/4*C*x^12+(27/11*B-162/11*C)*x^11+(27/10*A-81/5*B+162/5*C)*x^10+(-18*A+36 
*B-18*C)*x^9+(81/2*A-81/4*B-27*C)*x^8+(-162/7*A-216/7*B+216/7*C)*x^7+(-36* 
A+36*B+6*C)*x^6+(216/5*A+36/5*B-72/5*C)*x^5+(9*A-18*B)*x^4+(-24*A+8/3*C)*x 
^3+4*B*x^2+8*A*x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^3 \, dx=\frac {9}{4} \, C x^{12} + \frac {27}{11} \, {\left (B - 6 \, C\right )} x^{11} + \frac {27}{10} \, {\left (A - 6 \, B + 12 \, C\right )} x^{10} - 18 \, {\left (A - 2 \, B + C\right )} x^{9} + \frac {27}{4} \, {\left (6 \, A - 3 \, B - 4 \, C\right )} x^{8} - \frac {54}{7} \, {\left (3 \, A + 4 \, B - 4 \, C\right )} x^{7} - 6 \, {\left (6 \, A - 6 \, B - C\right )} x^{6} + \frac {36}{5} \, {\left (6 \, A + B - 2 \, C\right )} x^{5} + 9 \, {\left (A - 2 \, B\right )} x^{4} - \frac {8}{3} \, {\left (9 \, A - C\right )} x^{3} + 4 \, B x^{2} + 8 \, A x \] Input:

integrate((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^3,x, algorithm="fricas")
 

Output:

9/4*C*x^12 + 27/11*(B - 6*C)*x^11 + 27/10*(A - 6*B + 12*C)*x^10 - 18*(A - 
2*B + C)*x^9 + 27/4*(6*A - 3*B - 4*C)*x^8 - 54/7*(3*A + 4*B - 4*C)*x^7 - 6 
*(6*A - 6*B - C)*x^6 + 36/5*(6*A + B - 2*C)*x^5 + 9*(A - 2*B)*x^4 - 8/3*(9 
*A - C)*x^3 + 4*B*x^2 + 8*A*x
 

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.07 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^3 \, dx=8 A x + 4 B x^{2} + \frac {9 C x^{12}}{4} + x^{11} \cdot \left (\frac {27 B}{11} - \frac {162 C}{11}\right ) + x^{10} \cdot \left (\frac {27 A}{10} - \frac {81 B}{5} + \frac {162 C}{5}\right ) + x^{9} \left (- 18 A + 36 B - 18 C\right ) + x^{8} \cdot \left (\frac {81 A}{2} - \frac {81 B}{4} - 27 C\right ) + x^{7} \left (- \frac {162 A}{7} - \frac {216 B}{7} + \frac {216 C}{7}\right ) + x^{6} \left (- 36 A + 36 B + 6 C\right ) + x^{5} \cdot \left (\frac {216 A}{5} + \frac {36 B}{5} - \frac {72 C}{5}\right ) + x^{4} \cdot \left (9 A - 18 B\right ) + x^{3} \left (- 24 A + \frac {8 C}{3}\right ) \] Input:

integrate((C*x**2+B*x+A)*(3*x**3-6*x**2+2)**3,x)
 

Output:

8*A*x + 4*B*x**2 + 9*C*x**12/4 + x**11*(27*B/11 - 162*C/11) + x**10*(27*A/ 
10 - 81*B/5 + 162*C/5) + x**9*(-18*A + 36*B - 18*C) + x**8*(81*A/2 - 81*B/ 
4 - 27*C) + x**7*(-162*A/7 - 216*B/7 + 216*C/7) + x**6*(-36*A + 36*B + 6*C 
) + x**5*(216*A/5 + 36*B/5 - 72*C/5) + x**4*(9*A - 18*B) + x**3*(-24*A + 8 
*C/3)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^3 \, dx=\frac {9}{4} \, C x^{12} + \frac {27}{11} \, {\left (B - 6 \, C\right )} x^{11} + \frac {27}{10} \, {\left (A - 6 \, B + 12 \, C\right )} x^{10} - 18 \, {\left (A - 2 \, B + C\right )} x^{9} + \frac {27}{4} \, {\left (6 \, A - 3 \, B - 4 \, C\right )} x^{8} - \frac {54}{7} \, {\left (3 \, A + 4 \, B - 4 \, C\right )} x^{7} - 6 \, {\left (6 \, A - 6 \, B - C\right )} x^{6} + \frac {36}{5} \, {\left (6 \, A + B - 2 \, C\right )} x^{5} + 9 \, {\left (A - 2 \, B\right )} x^{4} - \frac {8}{3} \, {\left (9 \, A - C\right )} x^{3} + 4 \, B x^{2} + 8 \, A x \] Input:

integrate((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^3,x, algorithm="maxima")
 

Output:

9/4*C*x^12 + 27/11*(B - 6*C)*x^11 + 27/10*(A - 6*B + 12*C)*x^10 - 18*(A - 
2*B + C)*x^9 + 27/4*(6*A - 3*B - 4*C)*x^8 - 54/7*(3*A + 4*B - 4*C)*x^7 - 6 
*(6*A - 6*B - C)*x^6 + 36/5*(6*A + B - 2*C)*x^5 + 9*(A - 2*B)*x^4 - 8/3*(9 
*A - C)*x^3 + 4*B*x^2 + 8*A*x
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.11 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^3 \, dx=\frac {9}{4} \, C x^{12} + \frac {27}{11} \, B x^{11} - \frac {162}{11} \, C x^{11} + \frac {27}{10} \, A x^{10} - \frac {81}{5} \, B x^{10} + \frac {162}{5} \, C x^{10} - 18 \, A x^{9} + 36 \, B x^{9} - 18 \, C x^{9} + \frac {81}{2} \, A x^{8} - \frac {81}{4} \, B x^{8} - 27 \, C x^{8} - \frac {162}{7} \, A x^{7} - \frac {216}{7} \, B x^{7} + \frac {216}{7} \, C x^{7} - 36 \, A x^{6} + 36 \, B x^{6} + 6 \, C x^{6} + \frac {216}{5} \, A x^{5} + \frac {36}{5} \, B x^{5} - \frac {72}{5} \, C x^{5} + 9 \, A x^{4} - 18 \, B x^{4} - 24 \, A x^{3} + \frac {8}{3} \, C x^{3} + 4 \, B x^{2} + 8 \, A x \] Input:

integrate((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^3,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

9/4*C*x^12 + 27/11*B*x^11 - 162/11*C*x^11 + 27/10*A*x^10 - 81/5*B*x^10 + 1 
62/5*C*x^10 - 18*A*x^9 + 36*B*x^9 - 18*C*x^9 + 81/2*A*x^8 - 81/4*B*x^8 - 2 
7*C*x^8 - 162/7*A*x^7 - 216/7*B*x^7 + 216/7*C*x^7 - 36*A*x^6 + 36*B*x^6 + 
6*C*x^6 + 216/5*A*x^5 + 36/5*B*x^5 - 72/5*C*x^5 + 9*A*x^4 - 18*B*x^4 - 24* 
A*x^3 + 8/3*C*x^3 + 4*B*x^2 + 8*A*x
 

Mupad [B] (verification not implemented)

Time = 12.24 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.95 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^3 \, dx=\frac {9\,C\,x^{12}}{4}+\left (\frac {27\,B}{11}-\frac {162\,C}{11}\right )\,x^{11}+\left (\frac {27\,A}{10}-\frac {81\,B}{5}+\frac {162\,C}{5}\right )\,x^{10}+\left (36\,B-18\,A-18\,C\right )\,x^9+\left (\frac {81\,A}{2}-\frac {81\,B}{4}-27\,C\right )\,x^8+\left (\frac {216\,C}{7}-\frac {216\,B}{7}-\frac {162\,A}{7}\right )\,x^7+\left (36\,B-36\,A+6\,C\right )\,x^6+\left (\frac {216\,A}{5}+\frac {36\,B}{5}-\frac {72\,C}{5}\right )\,x^5+\left (9\,A-18\,B\right )\,x^4+\left (\frac {8\,C}{3}-24\,A\right )\,x^3+4\,B\,x^2+8\,A\,x \] Input:

int((A + B*x + C*x^2)*(3*x^3 - 6*x^2 + 2)^3,x)
 

Output:

8*A*x + 4*B*x^2 + (9*C*x^12)/4 - x^9*(18*A - 36*B + 18*C) + x^6*(36*B - 36 
*A + 6*C) - x^8*((81*B)/4 - (81*A)/2 + 27*C) + x^10*((27*A)/10 - (81*B)/5 
+ (162*C)/5) + x^5*((216*A)/5 + (36*B)/5 - (72*C)/5) - x^7*((162*A)/7 + (2 
16*B)/7 - (216*C)/7) + x^4*(9*A - 18*B) - x^3*(24*A - (8*C)/3) + x^11*((27 
*B)/11 - (162*C)/11)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.11 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^3 \, dx=\frac {x \left (10395 c \,x^{11}+11340 b \,x^{10}-68040 c \,x^{10}+12474 a \,x^{9}-74844 b \,x^{9}+149688 c \,x^{9}-83160 a \,x^{8}+166320 b \,x^{8}-83160 c \,x^{8}+187110 a \,x^{7}-93555 b \,x^{7}-124740 c \,x^{7}-106920 a \,x^{6}-142560 b \,x^{6}+142560 c \,x^{6}-166320 a \,x^{5}+166320 b \,x^{5}+27720 c \,x^{5}+199584 a \,x^{4}+33264 b \,x^{4}-66528 c \,x^{4}+41580 a \,x^{3}-83160 b \,x^{3}-110880 a \,x^{2}+12320 c \,x^{2}+18480 b x +36960 a \right )}{4620} \] Input:

int((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^3,x)
 

Output:

(x*(12474*a*x**9 - 83160*a*x**8 + 187110*a*x**7 - 106920*a*x**6 - 166320*a 
*x**5 + 199584*a*x**4 + 41580*a*x**3 - 110880*a*x**2 + 36960*a + 11340*b*x 
**10 - 74844*b*x**9 + 166320*b*x**8 - 93555*b*x**7 - 142560*b*x**6 + 16632 
0*b*x**5 + 33264*b*x**4 - 83160*b*x**3 + 18480*b*x + 10395*c*x**11 - 68040 
*c*x**10 + 149688*c*x**9 - 83160*c*x**8 - 124740*c*x**7 + 142560*c*x**6 + 
27720*c*x**5 - 66528*c*x**4 + 12320*c*x**2))/4620