Integrand size = 23, antiderivative size = 99 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^2 \, dx=4 A x-2 (6 A-B) x^2+\frac {4}{3} (9 A-6 B+C) x^3+3 (A+3 B-2 C) x^4-\frac {12}{5} (3 A-B-3 C) x^5-2 (3 B-C) x^6+\frac {9}{7} (A-4 C) x^7+\frac {9 B x^8}{8}+C x^9 \] Output:
4*A*x-2*(6*A-B)*x^2+4/3*(9*A-6*B+C)*x^3+3*(A+3*B-2*C)*x^4-12/5*(3*A-B-3*C) *x^5-2*(3*B-C)*x^6+9/7*(A-4*C)*x^7+9/8*B*x^8+C*x^9
Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^2 \, dx=4 A x-2 (6 A-B) x^2+\frac {4}{3} (9 A-6 B+C) x^3+3 (A+3 B-2 C) x^4-\frac {12}{5} (3 A-B-3 C) x^5-2 (3 B-C) x^6+\frac {9}{7} (A-4 C) x^7+\frac {9 B x^8}{8}+C x^9 \] Input:
Integrate[(A + B*x + C*x^2)*(2 - 6*x + 3*x^3)^2,x]
Output:
4*A*x - 2*(6*A - B)*x^2 + (4*(9*A - 6*B + C)*x^3)/3 + 3*(A + 3*B - 2*C)*x^ 4 - (12*(3*A - B - 3*C)*x^5)/5 - 2*(3*B - C)*x^6 + (9*(A - 4*C)*x^7)/7 + ( 9*B*x^8)/8 + C*x^9
Time = 0.54 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
\(\displaystyle \int \left (3 x^3-6 x+2\right )^2 \left (A+B x+C x^2\right ) \, dx\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \int \left (-12 x^4 (3 A-B-3 C)+12 x^3 (A+3 B-2 C)+4 x^2 (9 A-6 B+C)-4 x (6 A-B)+9 x^6 (A-4 C)+4 A-12 x^5 (3 B-C)+9 B x^7+9 C x^8\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {12}{5} x^5 (3 A-B-3 C)+3 x^4 (A+3 B-2 C)+\frac {4}{3} x^3 (9 A-6 B+C)-2 x^2 (6 A-B)+\frac {9}{7} x^7 (A-4 C)+4 A x-2 x^6 (3 B-C)+\frac {9 B x^8}{8}+C x^9\) |
Input:
Int[(A + B*x + C*x^2)*(2 - 6*x + 3*x^3)^2,x]
Output:
4*A*x - 2*(6*A - B)*x^2 + (4*(9*A - 6*B + C)*x^3)/3 + 3*(A + 3*B - 2*C)*x^ 4 - (12*(3*A - B - 3*C)*x^5)/5 - 2*(3*B - C)*x^6 + (9*(A - 4*C)*x^7)/7 + ( 9*B*x^8)/8 + C*x^9
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]
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.93
method | result | size |
norman | \(C \,x^{9}+\frac {9 B \,x^{8}}{8}+\left (\frac {9 A}{7}-\frac {36 C}{7}\right ) x^{7}+\left (-6 B +2 C \right ) x^{6}+\left (-\frac {36 A}{5}+\frac {12 B}{5}+\frac {36 C}{5}\right ) x^{5}+\left (3 A +9 B -6 C \right ) x^{4}+\left (12 A -8 B +\frac {4 C}{3}\right ) x^{3}+\left (-12 A +2 B \right ) x^{2}+4 A x\) | \(92\) |
default | \(C \,x^{9}+\frac {9 B \,x^{8}}{8}+\frac {\left (9 A -36 C \right ) x^{7}}{7}+\frac {\left (-36 B +12 C \right ) x^{6}}{6}+\frac {\left (-36 A +12 B +36 C \right ) x^{5}}{5}+\frac {\left (12 A +36 B -24 C \right ) x^{4}}{4}+\frac {\left (36 A -24 B +4 C \right ) x^{3}}{3}+\frac {\left (-24 A +4 B \right ) x^{2}}{2}+4 A x\) | \(98\) |
orering | \(\frac {x \left (840 x^{8} C +945 x^{7} B +1080 x^{6} A -4320 C \,x^{6}-5040 B \,x^{5}+1680 x^{5} C -6048 x^{4} A +2016 x^{4} B +6048 C \,x^{4}+2520 x^{3} A +7560 B \,x^{3}-5040 C \,x^{3}+10080 A \,x^{2}-6720 B \,x^{2}+1120 C \,x^{2}-10080 A x +1680 B x +3360 A \right )}{840}\) | \(106\) |
gosper | \(C \,x^{9}+\frac {9}{8} B \,x^{8}+\frac {9}{7} x^{7} A -\frac {36}{7} x^{7} C -6 x^{6} B +2 C \,x^{6}-\frac {36}{5} x^{5} A +\frac {12}{5} B \,x^{5}+\frac {36}{5} x^{5} C +3 x^{4} A +9 x^{4} B -6 C \,x^{4}+12 x^{3} A -8 B \,x^{3}+\frac {4}{3} C \,x^{3}-12 A \,x^{2}+2 B \,x^{2}+4 A x\) | \(107\) |
risch | \(C \,x^{9}+\frac {9}{8} B \,x^{8}+\frac {9}{7} x^{7} A -\frac {36}{7} x^{7} C -6 x^{6} B +2 C \,x^{6}-\frac {36}{5} x^{5} A +\frac {12}{5} B \,x^{5}+\frac {36}{5} x^{5} C +3 x^{4} A +9 x^{4} B -6 C \,x^{4}+12 x^{3} A -8 B \,x^{3}+\frac {4}{3} C \,x^{3}-12 A \,x^{2}+2 B \,x^{2}+4 A x\) | \(107\) |
parallelrisch | \(C \,x^{9}+\frac {9}{8} B \,x^{8}+\frac {9}{7} x^{7} A -\frac {36}{7} x^{7} C -6 x^{6} B +2 C \,x^{6}-\frac {36}{5} x^{5} A +\frac {12}{5} B \,x^{5}+\frac {36}{5} x^{5} C +3 x^{4} A +9 x^{4} B -6 C \,x^{4}+12 x^{3} A -8 B \,x^{3}+\frac {4}{3} C \,x^{3}-12 A \,x^{2}+2 B \,x^{2}+4 A x\) | \(107\) |
Input:
int((C*x^2+B*x+A)*(3*x^3-6*x+2)^2,x,method=_RETURNVERBOSE)
Output:
C*x^9+9/8*B*x^8+(9/7*A-36/7*C)*x^7+(-6*B+2*C)*x^6+(-36/5*A+12/5*B+36/5*C)* x^5+(3*A+9*B-6*C)*x^4+(12*A-8*B+4/3*C)*x^3+(-12*A+2*B)*x^2+4*A*x
Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.92 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^2 \, dx=C x^{9} + \frac {9}{8} \, B x^{8} + \frac {9}{7} \, {\left (A - 4 \, C\right )} x^{7} - 2 \, {\left (3 \, B - C\right )} x^{6} - \frac {12}{5} \, {\left (3 \, A - B - 3 \, C\right )} x^{5} + 3 \, {\left (A + 3 \, B - 2 \, C\right )} x^{4} + \frac {4}{3} \, {\left (9 \, A - 6 \, B + C\right )} x^{3} - 2 \, {\left (6 \, A - B\right )} x^{2} + 4 \, A x \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-6*x+2)^2,x, algorithm="fricas")
Output:
C*x^9 + 9/8*B*x^8 + 9/7*(A - 4*C)*x^7 - 2*(3*B - C)*x^6 - 12/5*(3*A - B - 3*C)*x^5 + 3*(A + 3*B - 2*C)*x^4 + 4/3*(9*A - 6*B + C)*x^3 - 2*(6*A - B)*x ^2 + 4*A*x
Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^2 \, dx=4 A x + \frac {9 B x^{8}}{8} + C x^{9} + x^{7} \cdot \left (\frac {9 A}{7} - \frac {36 C}{7}\right ) + x^{6} \left (- 6 B + 2 C\right ) + x^{5} \left (- \frac {36 A}{5} + \frac {12 B}{5} + \frac {36 C}{5}\right ) + x^{4} \cdot \left (3 A + 9 B - 6 C\right ) + x^{3} \cdot \left (12 A - 8 B + \frac {4 C}{3}\right ) + x^{2} \left (- 12 A + 2 B\right ) \] Input:
integrate((C*x**2+B*x+A)*(3*x**3-6*x+2)**2,x)
Output:
4*A*x + 9*B*x**8/8 + C*x**9 + x**7*(9*A/7 - 36*C/7) + x**6*(-6*B + 2*C) + x**5*(-36*A/5 + 12*B/5 + 36*C/5) + x**4*(3*A + 9*B - 6*C) + x**3*(12*A - 8 *B + 4*C/3) + x**2*(-12*A + 2*B)
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.92 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^2 \, dx=C x^{9} + \frac {9}{8} \, B x^{8} + \frac {9}{7} \, {\left (A - 4 \, C\right )} x^{7} - 2 \, {\left (3 \, B - C\right )} x^{6} - \frac {12}{5} \, {\left (3 \, A - B - 3 \, C\right )} x^{5} + 3 \, {\left (A + 3 \, B - 2 \, C\right )} x^{4} + \frac {4}{3} \, {\left (9 \, A - 6 \, B + C\right )} x^{3} - 2 \, {\left (6 \, A - B\right )} x^{2} + 4 \, A x \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-6*x+2)^2,x, algorithm="maxima")
Output:
C*x^9 + 9/8*B*x^8 + 9/7*(A - 4*C)*x^7 - 2*(3*B - C)*x^6 - 12/5*(3*A - B - 3*C)*x^5 + 3*(A + 3*B - 2*C)*x^4 + 4/3*(9*A - 6*B + C)*x^3 - 2*(6*A - B)*x ^2 + 4*A*x
Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.07 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^2 \, dx=C x^{9} + \frac {9}{8} \, B x^{8} + \frac {9}{7} \, A x^{7} - \frac {36}{7} \, C x^{7} - 6 \, B x^{6} + 2 \, C x^{6} - \frac {36}{5} \, A x^{5} + \frac {12}{5} \, B x^{5} + \frac {36}{5} \, C x^{5} + 3 \, A x^{4} + 9 \, B x^{4} - 6 \, C x^{4} + 12 \, A x^{3} - 8 \, B x^{3} + \frac {4}{3} \, C x^{3} - 12 \, A x^{2} + 2 \, B x^{2} + 4 \, A x \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-6*x+2)^2,x, algorithm="giac")
Output:
C*x^9 + 9/8*B*x^8 + 9/7*A*x^7 - 36/7*C*x^7 - 6*B*x^6 + 2*C*x^6 - 36/5*A*x^ 5 + 12/5*B*x^5 + 36/5*C*x^5 + 3*A*x^4 + 9*B*x^4 - 6*C*x^4 + 12*A*x^3 - 8*B *x^3 + 4/3*C*x^3 - 12*A*x^2 + 2*B*x^2 + 4*A*x
Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.94 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^2 \, dx=C\,x^9+\frac {9\,B\,x^8}{8}+\left (\frac {9\,A}{7}-\frac {36\,C}{7}\right )\,x^7+\left (2\,C-6\,B\right )\,x^6+\left (\frac {12\,B}{5}-\frac {36\,A}{5}+\frac {36\,C}{5}\right )\,x^5+\left (3\,A+9\,B-6\,C\right )\,x^4+\left (12\,A-8\,B+\frac {4\,C}{3}\right )\,x^3+\left (2\,B-12\,A\right )\,x^2+4\,A\,x \] Input:
int((A + B*x + C*x^2)*(3*x^3 - 6*x + 2)^2,x)
Output:
4*A*x + (9*B*x^8)/8 + C*x^9 + x^4*(3*A + 9*B - 6*C) + x^3*(12*A - 8*B + (4 *C)/3) + x^5*((12*B)/5 - (36*A)/5 + (36*C)/5) - x^2*(12*A - 2*B) + x^7*((9 *A)/7 - (36*C)/7) - x^6*(6*B - 2*C)
Time = 0.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.06 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x+3 x^3\right )^2 \, dx=\frac {x \left (840 c \,x^{8}+945 b \,x^{7}+1080 a \,x^{6}-4320 c \,x^{6}-5040 b \,x^{5}+1680 c \,x^{5}-6048 a \,x^{4}+2016 b \,x^{4}+6048 c \,x^{4}+2520 a \,x^{3}+7560 b \,x^{3}-5040 c \,x^{3}+10080 a \,x^{2}-6720 b \,x^{2}+1120 c \,x^{2}-10080 a x +1680 b x +3360 a \right )}{840} \] Input:
int((C*x^2+B*x+A)*(3*x^3-6*x+2)^2,x)
Output:
(x*(1080*a*x**6 - 6048*a*x**4 + 2520*a*x**3 + 10080*a*x**2 - 10080*a*x + 3 360*a + 945*b*x**7 - 5040*b*x**5 + 2016*b*x**4 + 7560*b*x**3 - 6720*b*x**2 + 1680*b*x + 840*c*x**8 - 4320*c*x**6 + 1680*c*x**5 + 6048*c*x**4 - 5040* c*x**3 + 1120*c*x**2))/840