Integrand size = 26, antiderivative size = 118 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^2 \, dx=4 A x+2 (3 A+B) x^2+\frac {1}{3} (25 A+12 B+4 C) x^3+\frac {1}{4} (28 A+25 B+12 C) x^4+\frac {1}{5} (22 A+28 B+25 C) x^5+\frac {1}{3} (4 A+11 B+14 C) x^6+\frac {1}{7} (A+8 B+22 C) x^7+\frac {1}{8} (B+8 C) x^8+\frac {C x^9}{9} \] Output:
4*A*x+2*(3*A+B)*x^2+1/3*(25*A+12*B+4*C)*x^3+1/4*(28*A+25*B+12*C)*x^4+1/5*( 22*A+28*B+25*C)*x^5+1/3*(4*A+11*B+14*C)*x^6+1/7*(A+8*B+22*C)*x^7+1/8*(B+8* C)*x^8+1/9*C*x^9
Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^2 \, dx=4 A x+2 (3 A+B) x^2+\frac {1}{3} (25 A+12 B+4 C) x^3+\frac {1}{4} (28 A+25 B+12 C) x^4+\frac {1}{5} (22 A+28 B+25 C) x^5+\frac {1}{3} (4 A+11 B+14 C) x^6+\frac {1}{7} (A+8 B+22 C) x^7+\frac {1}{8} (B+8 C) x^8+\frac {C x^9}{9} \] Input:
Integrate[(A + B*x + C*x^2)*(2 + 3*x + 4*x^2 + x^3)^2,x]
Output:
4*A*x + 2*(3*A + B)*x^2 + ((25*A + 12*B + 4*C)*x^3)/3 + ((28*A + 25*B + 12 *C)*x^4)/4 + ((22*A + 28*B + 25*C)*x^5)/5 + ((4*A + 11*B + 14*C)*x^6)/3 + ((A + 8*B + 22*C)*x^7)/7 + ((B + 8*C)*x^8)/8 + (C*x^9)/9
Time = 0.32 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
\(\displaystyle \int \left (x^3+4 x^2+3 x+2\right )^2 \left (A+B x+C x^2\right ) \, dx\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \int \left (x^6 (A+8 B+22 C)+2 x^5 (4 A+11 B+14 C)+x^4 (22 A+28 B+25 C)+x^3 (28 A+25 B+12 C)+x^2 (25 A+12 B+4 C)+4 x (3 A+B)+4 A+x^7 (B+8 C)+C x^8\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{7} x^7 (A+8 B+22 C)+\frac {1}{3} x^6 (4 A+11 B+14 C)+\frac {1}{5} x^5 (22 A+28 B+25 C)+\frac {1}{4} x^4 (28 A+25 B+12 C)+\frac {1}{3} x^3 (25 A+12 B+4 C)+2 x^2 (3 A+B)+4 A x+\frac {1}{8} x^8 (B+8 C)+\frac {C x^9}{9}\) |
Input:
Int[(A + B*x + C*x^2)*(2 + 3*x + 4*x^2 + x^3)^2,x]
Output:
4*A*x + 2*(3*A + B)*x^2 + ((25*A + 12*B + 4*C)*x^3)/3 + ((28*A + 25*B + 12 *C)*x^4)/4 + ((22*A + 28*B + 25*C)*x^5)/5 + ((4*A + 11*B + 14*C)*x^6)/3 + ((A + 8*B + 22*C)*x^7)/7 + ((B + 8*C)*x^8)/8 + (C*x^9)/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 = 102, normalized size of antiderivative = 0.86
method | result | size |
norman | \(\frac {C \,x^{9}}{9}+\left (\frac {B}{8}+C \right ) x^{8}+\left (\frac {A}{7}+\frac {8 B}{7}+\frac {22 C}{7}\right ) x^{7}+\left (\frac {4 A}{3}+\frac {11 B}{3}+\frac {14 C}{3}\right ) x^{6}+\left (\frac {22 A}{5}+\frac {28 B}{5}+5 C \right ) x^{5}+\left (7 A +\frac {25 B}{4}+3 C \right ) x^{4}+\left (\frac {25 A}{3}+4 B +\frac {4 C}{3}\right ) x^{3}+\left (6 A +2 B \right ) x^{2}+4 A x\) | \(102\) |
default | \(\frac {C \,x^{9}}{9}+\frac {\left (B +8 C \right ) x^{8}}{8}+\frac {\left (A +8 B +22 C \right ) x^{7}}{7}+\frac {\left (8 A +22 B +28 C \right ) x^{6}}{6}+\frac {\left (22 A +28 B +25 C \right ) x^{5}}{5}+\frac {\left (28 A +25 B +12 C \right ) x^{4}}{4}+\frac {\left (25 A +12 B +4 C \right ) x^{3}}{3}+\frac {\left (12 A +4 B \right ) x^{2}}{2}+4 A x\) | \(107\) |
orering | \(\frac {x \left (280 x^{8} C +315 x^{7} B +2520 x^{7} C +360 x^{6} A +2880 x^{6} B +7920 C \,x^{6}+3360 x^{5} A +9240 B \,x^{5}+11760 x^{5} C +11088 x^{4} A +14112 x^{4} B +12600 C \,x^{4}+17640 x^{3} A +15750 B \,x^{3}+7560 C \,x^{3}+21000 A \,x^{2}+10080 B \,x^{2}+3360 C \,x^{2}+15120 A x +5040 B x +10080 A \right )}{2520}\) | \(124\) |
gosper | \(\frac {1}{9} C \,x^{9}+\frac {1}{8} B \,x^{8}+x^{8} C +\frac {1}{7} x^{7} A +\frac {8}{7} x^{7} B +\frac {22}{7} x^{7} C +\frac {4}{3} x^{6} A +\frac {11}{3} x^{6} B +\frac {14}{3} C \,x^{6}+\frac {22}{5} x^{5} A +\frac {28}{5} B \,x^{5}+5 x^{5} C +7 x^{4} A +\frac {25}{4} x^{4} B +3 C \,x^{4}+\frac {25}{3} x^{3} A +4 B \,x^{3}+\frac {4}{3} C \,x^{3}+6 A \,x^{2}+2 B \,x^{2}+4 A x\) | \(125\) |
risch | \(\frac {1}{9} C \,x^{9}+\frac {1}{8} B \,x^{8}+x^{8} C +\frac {1}{7} x^{7} A +\frac {8}{7} x^{7} B +\frac {22}{7} x^{7} C +\frac {4}{3} x^{6} A +\frac {11}{3} x^{6} B +\frac {14}{3} C \,x^{6}+\frac {22}{5} x^{5} A +\frac {28}{5} B \,x^{5}+5 x^{5} C +7 x^{4} A +\frac {25}{4} x^{4} B +3 C \,x^{4}+\frac {25}{3} x^{3} A +4 B \,x^{3}+\frac {4}{3} C \,x^{3}+6 A \,x^{2}+2 B \,x^{2}+4 A x\) | \(125\) |
parallelrisch | \(\frac {1}{9} C \,x^{9}+\frac {1}{8} B \,x^{8}+x^{8} C +\frac {1}{7} x^{7} A +\frac {8}{7} x^{7} B +\frac {22}{7} x^{7} C +\frac {4}{3} x^{6} A +\frac {11}{3} x^{6} B +\frac {14}{3} C \,x^{6}+\frac {22}{5} x^{5} A +\frac {28}{5} B \,x^{5}+5 x^{5} C +7 x^{4} A +\frac {25}{4} x^{4} B +3 C \,x^{4}+\frac {25}{3} x^{3} A +4 B \,x^{3}+\frac {4}{3} C \,x^{3}+6 A \,x^{2}+2 B \,x^{2}+4 A x\) | \(125\) |
Input:
int((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
1/9*C*x^9+(1/8*B+C)*x^8+(1/7*A+8/7*B+22/7*C)*x^7+(4/3*A+11/3*B+14/3*C)*x^6 +(22/5*A+28/5*B+5*C)*x^5+(7*A+25/4*B+3*C)*x^4+(25/3*A+4*B+4/3*C)*x^3+(6*A+ 2*B)*x^2+4*A*x
Time = 0.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.88 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^2 \, dx=\frac {1}{9} \, C x^{9} + \frac {1}{8} \, {\left (B + 8 \, C\right )} x^{8} + \frac {1}{7} \, {\left (A + 8 \, B + 22 \, C\right )} x^{7} + \frac {1}{3} \, {\left (4 \, A + 11 \, B + 14 \, C\right )} x^{6} + \frac {1}{5} \, {\left (22 \, A + 28 \, B + 25 \, C\right )} x^{5} + \frac {1}{4} \, {\left (28 \, A + 25 \, B + 12 \, C\right )} x^{4} + \frac {1}{3} \, {\left (25 \, A + 12 \, B + 4 \, C\right )} x^{3} + 2 \, {\left (3 \, A + B\right )} x^{2} + 4 \, A x \] Input:
integrate((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^2,x, algorithm="fricas")
Output:
1/9*C*x^9 + 1/8*(B + 8*C)*x^8 + 1/7*(A + 8*B + 22*C)*x^7 + 1/3*(4*A + 11*B + 14*C)*x^6 + 1/5*(22*A + 28*B + 25*C)*x^5 + 1/4*(28*A + 25*B + 12*C)*x^4 + 1/3*(25*A + 12*B + 4*C)*x^3 + 2*(3*A + B)*x^2 + 4*A*x
Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^2 \, dx=4 A x + \frac {C x^{9}}{9} + x^{8} \left (\frac {B}{8} + C\right ) + x^{7} \left (\frac {A}{7} + \frac {8 B}{7} + \frac {22 C}{7}\right ) + x^{6} \cdot \left (\frac {4 A}{3} + \frac {11 B}{3} + \frac {14 C}{3}\right ) + x^{5} \cdot \left (\frac {22 A}{5} + \frac {28 B}{5} + 5 C\right ) + x^{4} \cdot \left (7 A + \frac {25 B}{4} + 3 C\right ) + x^{3} \cdot \left (\frac {25 A}{3} + 4 B + \frac {4 C}{3}\right ) + x^{2} \cdot \left (6 A + 2 B\right ) \] Input:
integrate((C*x**2+B*x+A)*(x**3+4*x**2+3*x+2)**2,x)
Output:
4*A*x + C*x**9/9 + x**8*(B/8 + C) + x**7*(A/7 + 8*B/7 + 22*C/7) + x**6*(4* A/3 + 11*B/3 + 14*C/3) + x**5*(22*A/5 + 28*B/5 + 5*C) + x**4*(7*A + 25*B/4 + 3*C) + x**3*(25*A/3 + 4*B + 4*C/3) + x**2*(6*A + 2*B)
Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.88 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^2 \, dx=\frac {1}{9} \, C x^{9} + \frac {1}{8} \, {\left (B + 8 \, C\right )} x^{8} + \frac {1}{7} \, {\left (A + 8 \, B + 22 \, C\right )} x^{7} + \frac {1}{3} \, {\left (4 \, A + 11 \, B + 14 \, C\right )} x^{6} + \frac {1}{5} \, {\left (22 \, A + 28 \, B + 25 \, C\right )} x^{5} + \frac {1}{4} \, {\left (28 \, A + 25 \, B + 12 \, C\right )} x^{4} + \frac {1}{3} \, {\left (25 \, A + 12 \, B + 4 \, C\right )} x^{3} + 2 \, {\left (3 \, A + B\right )} x^{2} + 4 \, A x \] Input:
integrate((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^2,x, algorithm="maxima")
Output:
1/9*C*x^9 + 1/8*(B + 8*C)*x^8 + 1/7*(A + 8*B + 22*C)*x^7 + 1/3*(4*A + 11*B + 14*C)*x^6 + 1/5*(22*A + 28*B + 25*C)*x^5 + 1/4*(28*A + 25*B + 12*C)*x^4 + 1/3*(25*A + 12*B + 4*C)*x^3 + 2*(3*A + B)*x^2 + 4*A*x
Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.05 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^2 \, dx=\frac {1}{9} \, C x^{9} + \frac {1}{8} \, B x^{8} + C x^{8} + \frac {1}{7} \, A x^{7} + \frac {8}{7} \, B x^{7} + \frac {22}{7} \, C x^{7} + \frac {4}{3} \, A x^{6} + \frac {11}{3} \, B x^{6} + \frac {14}{3} \, C x^{6} + \frac {22}{5} \, A x^{5} + \frac {28}{5} \, B x^{5} + 5 \, C x^{5} + 7 \, A x^{4} + \frac {25}{4} \, B x^{4} + 3 \, C x^{4} + \frac {25}{3} \, A x^{3} + 4 \, B x^{3} + \frac {4}{3} \, C x^{3} + 6 \, A x^{2} + 2 \, B x^{2} + 4 \, A x \] Input:
integrate((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^2,x, algorithm="giac")
Output:
1/9*C*x^9 + 1/8*B*x^8 + C*x^8 + 1/7*A*x^7 + 8/7*B*x^7 + 22/7*C*x^7 + 4/3*A *x^6 + 11/3*B*x^6 + 14/3*C*x^6 + 22/5*A*x^5 + 28/5*B*x^5 + 5*C*x^5 + 7*A*x ^4 + 25/4*B*x^4 + 3*C*x^4 + 25/3*A*x^3 + 4*B*x^3 + 4/3*C*x^3 + 6*A*x^2 + 2 *B*x^2 + 4*A*x
Time = 0.06 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.86 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^2 \, dx=\frac {C\,x^9}{9}+\left (\frac {B}{8}+C\right )\,x^8+\left (\frac {A}{7}+\frac {8\,B}{7}+\frac {22\,C}{7}\right )\,x^7+\left (\frac {4\,A}{3}+\frac {11\,B}{3}+\frac {14\,C}{3}\right )\,x^6+\left (\frac {22\,A}{5}+\frac {28\,B}{5}+5\,C\right )\,x^5+\left (7\,A+\frac {25\,B}{4}+3\,C\right )\,x^4+\left (\frac {25\,A}{3}+4\,B+\frac {4\,C}{3}\right )\,x^3+\left (6\,A+2\,B\right )\,x^2+4\,A\,x \] Input:
int((A + B*x + C*x^2)*(3*x + 4*x^2 + x^3 + 2)^2,x)
Output:
4*A*x + (C*x^9)/9 + x^3*((25*A)/3 + 4*B + (4*C)/3) + x^4*(7*A + (25*B)/4 + 3*C) + x^6*((4*A)/3 + (11*B)/3 + (14*C)/3) + x^7*(A/7 + (8*B)/7 + (22*C)/ 7) + x^5*((22*A)/5 + (28*B)/5 + 5*C) + x^8*(B/8 + C) + x^2*(6*A + 2*B)
Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.04 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^2 \, dx=\frac {x \left (280 c \,x^{8}+315 b \,x^{7}+2520 c \,x^{7}+360 a \,x^{6}+2880 b \,x^{6}+7920 c \,x^{6}+3360 a \,x^{5}+9240 b \,x^{5}+11760 c \,x^{5}+11088 a \,x^{4}+14112 b \,x^{4}+12600 c \,x^{4}+17640 a \,x^{3}+15750 b \,x^{3}+7560 c \,x^{3}+21000 a \,x^{2}+10080 b \,x^{2}+3360 c \,x^{2}+15120 a x +5040 b x +10080 a \right )}{2520} \] Input:
int((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^2,x)
Output:
(x*(360*a*x**6 + 3360*a*x**5 + 11088*a*x**4 + 17640*a*x**3 + 21000*a*x**2 + 15120*a*x + 10080*a + 315*b*x**7 + 2880*b*x**6 + 9240*b*x**5 + 14112*b*x **4 + 15750*b*x**3 + 10080*b*x**2 + 5040*b*x + 280*c*x**8 + 2520*c*x**7 + 7920*c*x**6 + 11760*c*x**5 + 12600*c*x**4 + 7560*c*x**3 + 3360*c*x**2))/25 20