Integrand size = 25, antiderivative size = 97 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^2 \, dx=4 A x+2 B x^2-\frac {4}{3} (6 A-C) x^3+3 (A-2 B) x^4+\frac {12}{5} (3 A+B-2 C) x^5-2 (3 A-3 B-C) x^6+\frac {9}{7} (A-4 B+4 C) x^7+\frac {9}{8} (B-4 C) x^8+C x^9 \] Output:
4*A*x+2*B*x^2-4/3*(6*A-C)*x^3+3*(A-2*B)*x^4+12/5*(3*A+B-2*C)*x^5-2*(3*A-3* B-C)*x^6+9/7*(A-4*B+4*C)*x^7+9/8*(B-4*C)*x^8+C*x^9
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^2 \, dx=4 A x+2 B x^2-\frac {4}{3} (6 A-C) x^3+3 (A-2 B) x^4+\frac {12}{5} (3 A+B-2 C) x^5-2 (3 A-3 B-C) x^6+\frac {9}{7} (A-4 B+4 C) x^7+\frac {9}{8} (B-4 C) x^8+C x^9 \] Input:
Integrate[(A + B*x + C*x^2)*(2 - 6*x^2 + 3*x^3)^2,x]
Output:
4*A*x + 2*B*x^2 - (4*(6*A - C)*x^3)/3 + 3*(A - 2*B)*x^4 + (12*(3*A + B - 2 *C)*x^5)/5 - 2*(3*A - 3*B - C)*x^6 + (9*(A - 4*B + 4*C)*x^7)/7 + (9*(B - 4 *C)*x^8)/8 + C*x^9
Time = 0.54 (sec) , antiderivative size = 97, 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.
\(\displaystyle \int \left (3 x^3-6 x^2+2\right )^2 \left (A+B x+C x^2\right ) \, dx\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \int \left (9 x^6 (A-4 B+4 C)-12 x^5 (3 A-3 B-C)+12 x^4 (3 A+B-2 C)+12 x^3 (A-2 B)-4 x^2 (6 A-C)+4 A+9 x^7 (B-4 C)+4 B x+9 C x^8\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {9}{7} x^7 (A-4 B+4 C)-2 x^6 (3 A-3 B-C)+\frac {12}{5} x^5 (3 A+B-2 C)+3 x^4 (A-2 B)-\frac {4}{3} x^3 (6 A-C)+4 A x+\frac {9}{8} x^8 (B-4 C)+2 B x^2+C x^9\) |
Input:
Int[(A + B*x + C*x^2)*(2 - 6*x^2 + 3*x^3)^2,x]
Output:
4*A*x + 2*B*x^2 - (4*(6*A - C)*x^3)/3 + 3*(A - 2*B)*x^4 + (12*(3*A + B - 2 *C)*x^5)/5 - 2*(3*A - 3*B - C)*x^6 + (9*(A - 4*B + 4*C)*x^7)/7 + (9*(B - 4 *C)*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 = 0.17 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95
method | result | size |
norman | \(C \,x^{9}+\left (\frac {9 B}{8}-\frac {9 C}{2}\right ) x^{8}+\left (\frac {9 A}{7}-\frac {36 B}{7}+\frac {36 C}{7}\right ) x^{7}+\left (-6 A +6 B +2 C \right ) x^{6}+\left (\frac {36 A}{5}+\frac {12 B}{5}-\frac {24 C}{5}\right ) x^{5}+\left (3 A -6 B \right ) x^{4}+\left (-8 A +\frac {4 C}{3}\right ) x^{3}+2 B \,x^{2}+4 A x\) | \(92\) |
default | \(C \,x^{9}+\frac {\left (9 B -36 C \right ) x^{8}}{8}+\frac {\left (9 A -36 B +36 C \right ) x^{7}}{7}+\frac {\left (-36 A +36 B +12 C \right ) x^{6}}{6}+\frac {\left (36 A +12 B -24 C \right ) x^{5}}{5}+\frac {\left (12 A -24 B \right ) x^{4}}{4}+\frac {\left (-24 A +4 C \right ) x^{3}}{3}+2 B \,x^{2}+4 A x\) | \(98\) |
gosper | \(C \,x^{9}+\frac {9}{8} B \,x^{8}-\frac {9}{2} x^{8} C +\frac {9}{7} x^{7} A -\frac {36}{7} x^{7} B +\frac {36}{7} x^{7} C -6 x^{6} A +6 x^{6} B +2 C \,x^{6}+\frac {36}{5} x^{5} A +\frac {12}{5} B \,x^{5}-\frac {24}{5} x^{5} C +3 x^{4} A -6 x^{4} B -8 x^{3} A +\frac {4}{3} C \,x^{3}+2 B \,x^{2}+4 A x\) | \(107\) |
risch | \(C \,x^{9}+\frac {9}{8} B \,x^{8}-\frac {9}{2} x^{8} C +\frac {9}{7} x^{7} A -\frac {36}{7} x^{7} B +\frac {36}{7} x^{7} C -6 x^{6} A +6 x^{6} B +2 C \,x^{6}+\frac {36}{5} x^{5} A +\frac {12}{5} B \,x^{5}-\frac {24}{5} x^{5} C +3 x^{4} A -6 x^{4} B -8 x^{3} A +\frac {4}{3} C \,x^{3}+2 B \,x^{2}+4 A x\) | \(107\) |
parallelrisch | \(C \,x^{9}+\frac {9}{8} B \,x^{8}-\frac {9}{2} x^{8} C +\frac {9}{7} x^{7} A -\frac {36}{7} x^{7} B +\frac {36}{7} x^{7} C -6 x^{6} A +6 x^{6} B +2 C \,x^{6}+\frac {36}{5} x^{5} A +\frac {12}{5} B \,x^{5}-\frac {24}{5} x^{5} C +3 x^{4} A -6 x^{4} B -8 x^{3} A +\frac {4}{3} C \,x^{3}+2 B \,x^{2}+4 A x\) | \(107\) |
orering | \(\frac {x \left (840 x^{8} C +945 x^{7} B -3780 x^{7} C +1080 x^{6} A -4320 x^{6} B +4320 C \,x^{6}-5040 x^{5} A +5040 B \,x^{5}+1680 x^{5} C +6048 x^{4} A +2016 x^{4} B -4032 C \,x^{4}+2520 x^{3} A -5040 B \,x^{3}-6720 A \,x^{2}+1120 C \,x^{2}+1680 B x +3360 A \right )}{840}\) | \(108\) |
Input:
int((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^2,x,method=_RETURNVERBOSE)
Output:
C*x^9+(9/8*B-9/2*C)*x^8+(9/7*A-36/7*B+36/7*C)*x^7+(-6*A+6*B+2*C)*x^6+(36/5 *A+12/5*B-24/5*C)*x^5+(3*A-6*B)*x^4+(-8*A+4/3*C)*x^3+2*B*x^2+4*A*x
Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.92 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^2 \, dx=C x^{9} + \frac {9}{8} \, {\left (B - 4 \, C\right )} x^{8} + \frac {9}{7} \, {\left (A - 4 \, B + 4 \, C\right )} x^{7} - 2 \, {\left (3 \, A - 3 \, B - C\right )} x^{6} + \frac {12}{5} \, {\left (3 \, A + B - 2 \, C\right )} x^{5} + 3 \, {\left (A - 2 \, B\right )} x^{4} - \frac {4}{3} \, {\left (6 \, A - C\right )} x^{3} + 2 \, B x^{2} + 4 \, A x \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^2,x, algorithm="fricas")
Output:
C*x^9 + 9/8*(B - 4*C)*x^8 + 9/7*(A - 4*B + 4*C)*x^7 - 2*(3*A - 3*B - C)*x^ 6 + 12/5*(3*A + B - 2*C)*x^5 + 3*(A - 2*B)*x^4 - 4/3*(6*A - C)*x^3 + 2*B*x ^2 + 4*A*x
Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.05 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^2 \, dx=4 A x + 2 B x^{2} + C x^{9} + x^{8} \cdot \left (\frac {9 B}{8} - \frac {9 C}{2}\right ) + x^{7} \cdot \left (\frac {9 A}{7} - \frac {36 B}{7} + \frac {36 C}{7}\right ) + x^{6} \left (- 6 A + 6 B + 2 C\right ) + x^{5} \cdot \left (\frac {36 A}{5} + \frac {12 B}{5} - \frac {24 C}{5}\right ) + x^{4} \cdot \left (3 A - 6 B\right ) + x^{3} \left (- 8 A + \frac {4 C}{3}\right ) \] Input:
integrate((C*x**2+B*x+A)*(3*x**3-6*x**2+2)**2,x)
Output:
4*A*x + 2*B*x**2 + C*x**9 + x**8*(9*B/8 - 9*C/2) + x**7*(9*A/7 - 36*B/7 + 36*C/7) + x**6*(-6*A + 6*B + 2*C) + x**5*(36*A/5 + 12*B/5 - 24*C/5) + x**4 *(3*A - 6*B) + x**3*(-8*A + 4*C/3)
Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.92 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^2 \, dx=C x^{9} + \frac {9}{8} \, {\left (B - 4 \, C\right )} x^{8} + \frac {9}{7} \, {\left (A - 4 \, B + 4 \, C\right )} x^{7} - 2 \, {\left (3 \, A - 3 \, B - C\right )} x^{6} + \frac {12}{5} \, {\left (3 \, A + B - 2 \, C\right )} x^{5} + 3 \, {\left (A - 2 \, B\right )} x^{4} - \frac {4}{3} \, {\left (6 \, A - C\right )} x^{3} + 2 \, B x^{2} + 4 \, A x \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^2,x, algorithm="maxima")
Output:
C*x^9 + 9/8*(B - 4*C)*x^8 + 9/7*(A - 4*B + 4*C)*x^7 - 2*(3*A - 3*B - C)*x^ 6 + 12/5*(3*A + B - 2*C)*x^5 + 3*(A - 2*B)*x^4 - 4/3*(6*A - C)*x^3 + 2*B*x ^2 + 4*A*x
Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.09 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^2 \, dx=C x^{9} + \frac {9}{8} \, B x^{8} - \frac {9}{2} \, C x^{8} + \frac {9}{7} \, A x^{7} - \frac {36}{7} \, B x^{7} + \frac {36}{7} \, C x^{7} - 6 \, A x^{6} + 6 \, B x^{6} + 2 \, C x^{6} + \frac {36}{5} \, A x^{5} + \frac {12}{5} \, B x^{5} - \frac {24}{5} \, C x^{5} + 3 \, A x^{4} - 6 \, B x^{4} - 8 \, A x^{3} + \frac {4}{3} \, C x^{3} + 2 \, B x^{2} + 4 \, A x \] Input:
integrate((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^2,x, algorithm="giac")
Output:
C*x^9 + 9/8*B*x^8 - 9/2*C*x^8 + 9/7*A*x^7 - 36/7*B*x^7 + 36/7*C*x^7 - 6*A* x^6 + 6*B*x^6 + 2*C*x^6 + 36/5*A*x^5 + 12/5*B*x^5 - 24/5*C*x^5 + 3*A*x^4 - 6*B*x^4 - 8*A*x^3 + 4/3*C*x^3 + 2*B*x^2 + 4*A*x
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^2 \, dx=C\,x^9+\left (\frac {9\,B}{8}-\frac {9\,C}{2}\right )\,x^8+\left (\frac {9\,A}{7}-\frac {36\,B}{7}+\frac {36\,C}{7}\right )\,x^7+\left (6\,B-6\,A+2\,C\right )\,x^6+\left (\frac {36\,A}{5}+\frac {12\,B}{5}-\frac {24\,C}{5}\right )\,x^5+\left (3\,A-6\,B\right )\,x^4+\left (\frac {4\,C}{3}-8\,A\right )\,x^3+2\,B\,x^2+4\,A\,x \] Input:
int((A + B*x + C*x^2)*(3*x^3 - 6*x^2 + 2)^2,x)
Output:
4*A*x + 2*B*x^2 + C*x^9 + x^6*(6*B - 6*A + 2*C) + x^5*((36*A)/5 + (12*B)/5 - (24*C)/5) + x^7*((9*A)/7 - (36*B)/7 + (36*C)/7) + x^4*(3*A - 6*B) - x^3 *(8*A - (4*C)/3) + x^8*((9*B)/8 - (9*C)/2)
Time = 0.17 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.10 \[ \int \left (A+B x+C x^2\right ) \left (2-6 x^2+3 x^3\right )^2 \, dx=\frac {x \left (840 c \,x^{8}+945 b \,x^{7}-3780 c \,x^{7}+1080 a \,x^{6}-4320 b \,x^{6}+4320 c \,x^{6}-5040 a \,x^{5}+5040 b \,x^{5}+1680 c \,x^{5}+6048 a \,x^{4}+2016 b \,x^{4}-4032 c \,x^{4}+2520 a \,x^{3}-5040 b \,x^{3}-6720 a \,x^{2}+1120 c \,x^{2}+1680 b x +3360 a \right )}{840} \] Input:
int((C*x^2+B*x+A)*(3*x^3-6*x^2+2)^2,x)
Output:
(x*(1080*a*x**6 - 5040*a*x**5 + 6048*a*x**4 + 2520*a*x**3 - 6720*a*x**2 + 3360*a + 945*b*x**7 - 4320*b*x**6 + 5040*b*x**5 + 2016*b*x**4 - 5040*b*x** 3 + 1680*b*x + 840*c*x**8 - 3780*c*x**7 + 4320*c*x**6 + 1680*c*x**5 - 4032 *c*x**4 + 1120*c*x**2))/840