Integrand size = 20, antiderivative size = 272 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{4 e^7}-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}{5 e^7}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^6}{2 e^7}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^7}{7 e^7}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^8}{8 e^7}-\frac {c^2 (2 c d-b e) (d+e x)^9}{3 e^7}+\frac {c^3 (d+e x)^{10}}{10 e^7} \] Output:
1/4*(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^4/e^7-3/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^ 2)^2*(e*x+d)^5/e^7+1/2*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5* b*d))*(e*x+d)^6/e^7-1/7*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b *d))*(e*x+d)^7/e^7+3/8*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^8/e^ 7-1/3*c^2*(-b*e+2*c*d)*(e*x+d)^9/e^7+1/10*c^3*(e*x+d)^10/e^7
Time = 0.10 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.37 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=a^3 d^3 x+\frac {3}{2} a^2 d^2 (b d+a e) x^2+a d \left (b^2 d^2+3 a b d e+a \left (c d^2+a e^2\right )\right ) x^3+\frac {1}{4} \left (b^3 d^3+9 a b^2 d^2 e+a^2 e \left (9 c d^2+a e^2\right )+3 a b d \left (2 c d^2+3 a e^2\right )\right ) x^4+\frac {3}{5} \left (b^3 d^2 e+a b e \left (6 c d^2+a e^2\right )+a c d \left (c d^2+3 a e^2\right )+b^2 \left (c d^3+3 a d e^2\right )\right ) x^5+\frac {1}{2} \left (b^3 d e^2+a c e \left (3 c d^2+a e^2\right )+b c d \left (c d^2+6 a e^2\right )+b^2 \left (3 c d^2 e+a e^3\right )\right ) x^6+\frac {1}{7} \left (c^3 d^3+b^3 e^3+9 c^2 d e (b d+a e)+3 b c e^2 (3 b d+2 a e)\right ) x^7+\frac {3}{8} c e \left (c^2 d^2+b^2 e^2+c e (3 b d+a e)\right ) x^8+\frac {1}{3} c^2 e^2 (c d+b e) x^9+\frac {1}{10} c^3 e^3 x^{10} \] Input:
Integrate[(d + e*x)^3*(a + b*x + c*x^2)^3,x]
Output:
a^3*d^3*x + (3*a^2*d^2*(b*d + a*e)*x^2)/2 + a*d*(b^2*d^2 + 3*a*b*d*e + a*( c*d^2 + a*e^2))*x^3 + ((b^3*d^3 + 9*a*b^2*d^2*e + a^2*e*(9*c*d^2 + a*e^2) + 3*a*b*d*(2*c*d^2 + 3*a*e^2))*x^4)/4 + (3*(b^3*d^2*e + a*b*e*(6*c*d^2 + a *e^2) + a*c*d*(c*d^2 + 3*a*e^2) + b^2*(c*d^3 + 3*a*d*e^2))*x^5)/5 + ((b^3* d*e^2 + a*c*e*(3*c*d^2 + a*e^2) + b*c*d*(c*d^2 + 6*a*e^2) + b^2*(3*c*d^2*e + a*e^3))*x^6)/2 + ((c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(b*d + a*e) + 3*b*c*e^ 2*(3*b*d + 2*a*e))*x^7)/7 + (3*c*e*(c^2*d^2 + b^2*e^2 + c*e*(3*b*d + a*e)) *x^8)/8 + (c^2*e^2*(c*d + b*e)*x^9)/3 + (c^3*e^3*x^10)/10
Time = 0.61 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1140, 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.
| \(\displaystyle \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c (d+e x)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {(d+e x)^6 (2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^6}+\frac {3 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}+\frac {3 (d+e x)^4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^6}+\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )^3}{e^6}-\frac {3 c^2 (d+e x)^8 (2 c d-b e)}{e^6}+\frac {c^3 (d+e x)^9}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 c (d+e x)^8 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{8 e^7}-\frac {(d+e x)^7 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^7}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7}-\frac {3 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )^3}{4 e^7}-\frac {c^2 (d+e x)^9 (2 c d-b e)}{3 e^7}+\frac {c^3 (d+e x)^{10}}{10 e^7}\) |
Input:
Int[(d + e*x)^3*(a + b*x + c*x^2)^3,x]
Output:
((c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^4)/(4*e^7) - (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^5)/(5*e^7) + ((c*d^2 - b*d*e + a*e^2)*(5*c^2* d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^6)/(2*e^7) - ((2*c*d - b*e)*( 10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^7)/(7*e^7) + (3*c* (5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^8)/(8*e^7) - (c^2*(2*c *d - b*e)*(d + e*x)^9)/(3*e^7) + (c^3*(d + e*x)^10)/(10*e^7)
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 0.84 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.49
| method | result | size |
| norman | \(\frac {c^{3} e^{3} x^{10}}{10}+\left (\frac {1}{3} e^{3} b \,c^{2}+\frac {1}{3} d \,e^{2} c^{3}\right ) x^{9}+\left (\frac {3}{8} a \,c^{2} e^{3}+\frac {3}{8} b^{2} c \,e^{3}+\frac {9}{8} d \,e^{2} b \,c^{2}+\frac {3}{8} d^{2} e \,c^{3}\right ) x^{8}+\left (\frac {6}{7} a b c \,e^{3}+\frac {9}{7} d \,e^{2} a \,c^{2}+\frac {1}{7} b^{3} e^{3}+\frac {9}{7} d \,e^{2} b^{2} c +\frac {9}{7} d^{2} e b \,c^{2}+\frac {1}{7} d^{3} c^{3}\right ) x^{7}+\left (\frac {1}{2} a^{2} c \,e^{3}+\frac {1}{2} b^{2} e^{3} a +3 b d \,e^{2} a c +\frac {3}{2} a \,c^{2} d^{2} e +\frac {1}{2} b^{3} e^{2} d +\frac {3}{2} b^{2} c \,d^{2} e +\frac {1}{2} d^{3} b \,c^{2}\right ) x^{6}+\left (\frac {3}{5} a^{2} b \,e^{3}+\frac {9}{5} a^{2} c d \,e^{2}+\frac {9}{5} a \,b^{2} d \,e^{2}+\frac {18}{5} a b c \,d^{2} e +\frac {3}{5} d^{3} a \,c^{2}+\frac {3}{5} b^{3} d^{2} e +\frac {3}{5} b^{2} c \,d^{3}\right ) x^{5}+\left (\frac {1}{4} e^{3} a^{3}+\frac {9}{4} a^{2} b d \,e^{2}+\frac {9}{4} a^{2} c \,d^{2} e +\frac {9}{4} a \,b^{2} d^{2} e +\frac {3}{2} a b c \,d^{3}+\frac {1}{4} b^{3} d^{3}\right ) x^{4}+\left (d \,e^{2} a^{3}+3 d^{2} e \,a^{2} b +d^{3} c \,a^{2}+a \,b^{2} d^{3}\right ) x^{3}+\left (\frac {3}{2} d^{2} e \,a^{3}+\frac {3}{2} d^{3} a^{2} b \right ) x^{2}+a^{3} d^{3} x\) | \(406\) |
| gosper | \(\frac {9}{5} x^{5} a \,b^{2} d \,e^{2}+\frac {9}{4} x^{4} a^{2} b d \,e^{2}+\frac {9}{4} x^{4} a^{2} c \,d^{2} e +\frac {9}{4} x^{4} a \,b^{2} d^{2} e +\frac {3}{5} x^{5} a^{2} b \,e^{3}+\frac {3}{5} x^{5} d^{3} a \,c^{2}+\frac {3}{5} x^{5} b^{3} d^{2} e +\frac {18}{5} x^{5} a b c \,d^{2} e +\frac {1}{3} x^{9} e^{3} b \,c^{2}+\frac {3}{8} x^{8} b^{2} c \,e^{3}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {1}{7} c^{3} d^{3} x^{7}+\frac {3}{2} d^{2} e \,a^{3} x^{2}+\frac {9}{8} x^{8} d \,e^{2} b \,c^{2}+\frac {6}{7} x^{7} a b c \,e^{3}+\frac {9}{7} x^{7} d \,e^{2} a \,c^{2}+\frac {9}{7} x^{7} d \,e^{2} b^{2} c +\frac {3}{2} a^{2} b \,d^{3} x^{2}+a^{2} c \,d^{3} x^{3}+\frac {9}{7} x^{7} d^{2} e b \,c^{2}+\frac {3}{2} x^{6} a \,c^{2} d^{2} e +\frac {3}{2} x^{6} b^{2} c \,d^{2} e +3 x^{6} b d \,e^{2} a c +\frac {9}{5} x^{5} a^{2} c d \,e^{2}+3 a^{2} b \,d^{2} e \,x^{3}+\frac {1}{4} a^{3} e^{3} x^{4}+\frac {1}{10} c^{3} e^{3} x^{10}+\frac {3}{5} b^{2} c \,d^{3} x^{5}+\frac {1}{2} a^{2} c \,e^{3} x^{6}+\frac {3}{8} a \,c^{2} e^{3} x^{8}+\frac {1}{3} c^{3} d \,e^{2} x^{9}+\frac {1}{2} x^{6} b^{2} e^{3} a +\frac {1}{2} x^{6} b^{3} e^{2} d +\frac {1}{2} x^{6} d^{3} b \,c^{2}+a^{3} d \,e^{2} x^{3}+a \,b^{2} d^{3} x^{3}+\frac {3}{2} a b c \,d^{3} x^{4}+a^{3} d^{3} x +\frac {1}{4} b^{3} d^{3} x^{4}+\frac {1}{7} x^{7} b^{3} e^{3}\) | \(480\) |
| risch | \(\frac {9}{5} x^{5} a \,b^{2} d \,e^{2}+\frac {9}{4} x^{4} a^{2} b d \,e^{2}+\frac {9}{4} x^{4} a^{2} c \,d^{2} e +\frac {9}{4} x^{4} a \,b^{2} d^{2} e +\frac {3}{5} x^{5} a^{2} b \,e^{3}+\frac {3}{5} x^{5} d^{3} a \,c^{2}+\frac {3}{5} x^{5} b^{3} d^{2} e +\frac {18}{5} x^{5} a b c \,d^{2} e +\frac {1}{3} x^{9} e^{3} b \,c^{2}+\frac {3}{8} x^{8} b^{2} c \,e^{3}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {1}{7} c^{3} d^{3} x^{7}+\frac {3}{2} d^{2} e \,a^{3} x^{2}+\frac {9}{8} x^{8} d \,e^{2} b \,c^{2}+\frac {6}{7} x^{7} a b c \,e^{3}+\frac {9}{7} x^{7} d \,e^{2} a \,c^{2}+\frac {9}{7} x^{7} d \,e^{2} b^{2} c +\frac {3}{2} a^{2} b \,d^{3} x^{2}+a^{2} c \,d^{3} x^{3}+\frac {9}{7} x^{7} d^{2} e b \,c^{2}+\frac {3}{2} x^{6} a \,c^{2} d^{2} e +\frac {3}{2} x^{6} b^{2} c \,d^{2} e +3 x^{6} b d \,e^{2} a c +\frac {9}{5} x^{5} a^{2} c d \,e^{2}+3 a^{2} b \,d^{2} e \,x^{3}+\frac {1}{4} a^{3} e^{3} x^{4}+\frac {1}{10} c^{3} e^{3} x^{10}+\frac {3}{5} b^{2} c \,d^{3} x^{5}+\frac {1}{2} a^{2} c \,e^{3} x^{6}+\frac {3}{8} a \,c^{2} e^{3} x^{8}+\frac {1}{3} c^{3} d \,e^{2} x^{9}+\frac {1}{2} x^{6} b^{2} e^{3} a +\frac {1}{2} x^{6} b^{3} e^{2} d +\frac {1}{2} x^{6} d^{3} b \,c^{2}+a^{3} d \,e^{2} x^{3}+a \,b^{2} d^{3} x^{3}+\frac {3}{2} a b c \,d^{3} x^{4}+a^{3} d^{3} x +\frac {1}{4} b^{3} d^{3} x^{4}+\frac {1}{7} x^{7} b^{3} e^{3}\) | \(480\) |
| parallelrisch | \(\frac {9}{5} x^{5} a \,b^{2} d \,e^{2}+\frac {9}{4} x^{4} a^{2} b d \,e^{2}+\frac {9}{4} x^{4} a^{2} c \,d^{2} e +\frac {9}{4} x^{4} a \,b^{2} d^{2} e +\frac {3}{5} x^{5} a^{2} b \,e^{3}+\frac {3}{5} x^{5} d^{3} a \,c^{2}+\frac {3}{5} x^{5} b^{3} d^{2} e +\frac {18}{5} x^{5} a b c \,d^{2} e +\frac {1}{3} x^{9} e^{3} b \,c^{2}+\frac {3}{8} x^{8} b^{2} c \,e^{3}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {1}{7} c^{3} d^{3} x^{7}+\frac {3}{2} d^{2} e \,a^{3} x^{2}+\frac {9}{8} x^{8} d \,e^{2} b \,c^{2}+\frac {6}{7} x^{7} a b c \,e^{3}+\frac {9}{7} x^{7} d \,e^{2} a \,c^{2}+\frac {9}{7} x^{7} d \,e^{2} b^{2} c +\frac {3}{2} a^{2} b \,d^{3} x^{2}+a^{2} c \,d^{3} x^{3}+\frac {9}{7} x^{7} d^{2} e b \,c^{2}+\frac {3}{2} x^{6} a \,c^{2} d^{2} e +\frac {3}{2} x^{6} b^{2} c \,d^{2} e +3 x^{6} b d \,e^{2} a c +\frac {9}{5} x^{5} a^{2} c d \,e^{2}+3 a^{2} b \,d^{2} e \,x^{3}+\frac {1}{4} a^{3} e^{3} x^{4}+\frac {1}{10} c^{3} e^{3} x^{10}+\frac {3}{5} b^{2} c \,d^{3} x^{5}+\frac {1}{2} a^{2} c \,e^{3} x^{6}+\frac {3}{8} a \,c^{2} e^{3} x^{8}+\frac {1}{3} c^{3} d \,e^{2} x^{9}+\frac {1}{2} x^{6} b^{2} e^{3} a +\frac {1}{2} x^{6} b^{3} e^{2} d +\frac {1}{2} x^{6} d^{3} b \,c^{2}+a^{3} d \,e^{2} x^{3}+a \,b^{2} d^{3} x^{3}+\frac {3}{2} a b c \,d^{3} x^{4}+a^{3} d^{3} x +\frac {1}{4} b^{3} d^{3} x^{4}+\frac {1}{7} x^{7} b^{3} e^{3}\) | \(480\) |
| orering | \(\frac {x \left (84 c^{3} e^{3} x^{9}+280 b \,c^{2} e^{3} x^{8}+280 c^{3} d \,e^{2} x^{8}+315 a \,c^{2} e^{3} x^{7}+315 b^{2} c \,e^{3} x^{7}+945 b \,c^{2} d \,e^{2} x^{7}+315 c^{3} d^{2} e \,x^{7}+720 a b c \,e^{3} x^{6}+1080 a \,c^{2} d \,e^{2} x^{6}+120 b^{3} e^{3} x^{6}+1080 b^{2} c d \,e^{2} x^{6}+1080 b \,c^{2} d^{2} e \,x^{6}+120 c^{3} d^{3} x^{6}+420 a^{2} c \,e^{3} x^{5}+420 x^{5} b^{2} e^{3} a +2520 a b c d \,e^{2} x^{5}+1260 a \,c^{2} d^{2} e \,x^{5}+420 x^{5} d \,e^{2} b^{3}+1260 b^{2} c \,d^{2} e \,x^{5}+420 b \,c^{2} d^{3} x^{5}+504 x^{4} a^{2} b \,e^{3}+1512 a^{2} c d \,e^{2} x^{4}+1512 x^{4} a \,b^{2} d \,e^{2}+3024 a b c \,d^{2} e \,x^{4}+504 a \,c^{2} d^{3} x^{4}+504 x^{4} b^{3} d^{2} e +504 b^{2} c \,d^{3} x^{4}+210 x^{3} e^{3} a^{3}+1890 x^{3} a^{2} b d \,e^{2}+1890 a^{2} c \,d^{2} e \,x^{3}+1890 x^{3} a \,b^{2} d^{2} e +1260 a b c \,d^{3} x^{3}+210 x^{3} b^{3} d^{3}+840 a^{3} d \,e^{2} x^{2}+2520 a^{2} b \,d^{2} e \,x^{2}+840 a^{2} c \,d^{3} x^{2}+840 a \,b^{2} d^{3} x^{2}+1260 x \,d^{2} e \,a^{3}+1260 a^{2} b \,d^{3} x +840 a^{3} d^{3}\right )}{840}\) | \(482\) |
| default | \(\frac {c^{3} e^{3} x^{10}}{10}+\frac {\left (3 e^{3} b \,c^{2}+3 d \,e^{2} c^{3}\right ) x^{9}}{9}+\frac {\left (3 d^{2} e \,c^{3}+9 d \,e^{2} b \,c^{2}+e^{3} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{8}}{8}+\frac {\left (d^{3} c^{3}+9 d^{2} e b \,c^{2}+3 d \,e^{2} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+e^{3} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (3 d^{3} b \,c^{2}+3 d^{2} e \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+3 d \,e^{2} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+e^{3} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{6}}{6}+\frac {\left (d^{3} \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+3 d^{2} e \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+3 d \,e^{2} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+3 a^{2} b \,e^{3}\right ) x^{5}}{5}+\frac {\left (d^{3} \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+3 d^{2} e \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+9 a^{2} b d \,e^{2}+e^{3} a^{3}\right ) x^{4}}{4}+\frac {\left (d^{3} \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+9 d^{2} e \,a^{2} b +3 d \,e^{2} a^{3}\right ) x^{3}}{3}+\frac {\left (3 d^{2} e \,a^{3}+3 d^{3} a^{2} b \right ) x^{2}}{2}+a^{3} d^{3} x\) | \(495\) |
Input:
int((e*x+d)^3*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/10*c^3*e^3*x^10+(1/3*e^3*b*c^2+1/3*d*e^2*c^3)*x^9+(3/8*a*c^2*e^3+3/8*b^2 *c*e^3+9/8*d*e^2*b*c^2+3/8*d^2*e*c^3)*x^8+(6/7*a*b*c*e^3+9/7*d*e^2*a*c^2+1 /7*b^3*e^3+9/7*d*e^2*b^2*c+9/7*d^2*e*b*c^2+1/7*d^3*c^3)*x^7+(1/2*a^2*c*e^3 +1/2*b^2*e^3*a+3*b*d*e^2*a*c+3/2*a*c^2*d^2*e+1/2*b^3*e^2*d+3/2*b^2*c*d^2*e +1/2*d^3*b*c^2)*x^6+(3/5*a^2*b*e^3+9/5*a^2*c*d*e^2+9/5*a*b^2*d*e^2+18/5*a* b*c*d^2*e+3/5*d^3*a*c^2+3/5*b^3*d^2*e+3/5*b^2*c*d^3)*x^5+(1/4*e^3*a^3+9/4* a^2*b*d*e^2+9/4*a^2*c*d^2*e+9/4*a*b^2*d^2*e+3/2*a*b*c*d^3+1/4*b^3*d^3)*x^4 +(a^3*d*e^2+3*a^2*b*d^2*e+a^2*c*d^3+a*b^2*d^3)*x^3+(3/2*d^2*e*a^3+3/2*d^3* a^2*b)*x^2+a^3*d^3*x
Time = 0.07 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.35 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, {\left (c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{9} + \frac {3}{8} \, {\left (c^{3} d^{2} e + 3 \, b c^{2} d e^{2} + {\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, b c^{2} d^{2} e + 9 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} + {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x^{7} + a^{3} d^{3} x + \frac {1}{2} \, {\left (b c^{2} d^{3} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e + {\left (b^{3} + 6 \, a b c\right )} d e^{2} + {\left (a b^{2} + a^{2} c\right )} e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (a^{2} b e^{3} + {\left (b^{2} c + a c^{2}\right )} d^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e + 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (9 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{3} + 9 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e\right )} x^{4} + {\left (3 \, a^{2} b d^{2} e + a^{3} d e^{2} + {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b d^{3} + a^{3} d^{2} e\right )} x^{2} \] Input:
integrate((e*x+d)^3*(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
1/10*c^3*e^3*x^10 + 1/3*(c^3*d*e^2 + b*c^2*e^3)*x^9 + 3/8*(c^3*d^2*e + 3*b *c^2*d*e^2 + (b^2*c + a*c^2)*e^3)*x^8 + 1/7*(c^3*d^3 + 9*b*c^2*d^2*e + 9*( b^2*c + a*c^2)*d*e^2 + (b^3 + 6*a*b*c)*e^3)*x^7 + a^3*d^3*x + 1/2*(b*c^2*d ^3 + 3*(b^2*c + a*c^2)*d^2*e + (b^3 + 6*a*b*c)*d*e^2 + (a*b^2 + a^2*c)*e^3 )*x^6 + 3/5*(a^2*b*e^3 + (b^2*c + a*c^2)*d^3 + (b^3 + 6*a*b*c)*d^2*e + 3*( a*b^2 + a^2*c)*d*e^2)*x^5 + 1/4*(9*a^2*b*d*e^2 + a^3*e^3 + (b^3 + 6*a*b*c) *d^3 + 9*(a*b^2 + a^2*c)*d^2*e)*x^4 + (3*a^2*b*d^2*e + a^3*d*e^2 + (a*b^2 + a^2*c)*d^3)*x^3 + 3/2*(a^2*b*d^3 + a^3*d^2*e)*x^2
Time = 0.06 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.78 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=a^{3} d^{3} x + \frac {c^{3} e^{3} x^{10}}{10} + x^{9} \left (\frac {b c^{2} e^{3}}{3} + \frac {c^{3} d e^{2}}{3}\right ) + x^{8} \cdot \left (\frac {3 a c^{2} e^{3}}{8} + \frac {3 b^{2} c e^{3}}{8} + \frac {9 b c^{2} d e^{2}}{8} + \frac {3 c^{3} d^{2} e}{8}\right ) + x^{7} \cdot \left (\frac {6 a b c e^{3}}{7} + \frac {9 a c^{2} d e^{2}}{7} + \frac {b^{3} e^{3}}{7} + \frac {9 b^{2} c d e^{2}}{7} + \frac {9 b c^{2} d^{2} e}{7} + \frac {c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac {a^{2} c e^{3}}{2} + \frac {a b^{2} e^{3}}{2} + 3 a b c d e^{2} + \frac {3 a c^{2} d^{2} e}{2} + \frac {b^{3} d e^{2}}{2} + \frac {3 b^{2} c d^{2} e}{2} + \frac {b c^{2} d^{3}}{2}\right ) + x^{5} \cdot \left (\frac {3 a^{2} b e^{3}}{5} + \frac {9 a^{2} c d e^{2}}{5} + \frac {9 a b^{2} d e^{2}}{5} + \frac {18 a b c d^{2} e}{5} + \frac {3 a c^{2} d^{3}}{5} + \frac {3 b^{3} d^{2} e}{5} + \frac {3 b^{2} c d^{3}}{5}\right ) + x^{4} \left (\frac {a^{3} e^{3}}{4} + \frac {9 a^{2} b d e^{2}}{4} + \frac {9 a^{2} c d^{2} e}{4} + \frac {9 a b^{2} d^{2} e}{4} + \frac {3 a b c d^{3}}{2} + \frac {b^{3} d^{3}}{4}\right ) + x^{3} \left (a^{3} d e^{2} + 3 a^{2} b d^{2} e + a^{2} c d^{3} + a b^{2} d^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{3} d^{2} e}{2} + \frac {3 a^{2} b d^{3}}{2}\right ) \] Input:
integrate((e*x+d)**3*(c*x**2+b*x+a)**3,x)
Output:
a**3*d**3*x + c**3*e**3*x**10/10 + x**9*(b*c**2*e**3/3 + c**3*d*e**2/3) + x**8*(3*a*c**2*e**3/8 + 3*b**2*c*e**3/8 + 9*b*c**2*d*e**2/8 + 3*c**3*d**2* e/8) + x**7*(6*a*b*c*e**3/7 + 9*a*c**2*d*e**2/7 + b**3*e**3/7 + 9*b**2*c*d *e**2/7 + 9*b*c**2*d**2*e/7 + c**3*d**3/7) + x**6*(a**2*c*e**3/2 + a*b**2* e**3/2 + 3*a*b*c*d*e**2 + 3*a*c**2*d**2*e/2 + b**3*d*e**2/2 + 3*b**2*c*d** 2*e/2 + b*c**2*d**3/2) + x**5*(3*a**2*b*e**3/5 + 9*a**2*c*d*e**2/5 + 9*a*b **2*d*e**2/5 + 18*a*b*c*d**2*e/5 + 3*a*c**2*d**3/5 + 3*b**3*d**2*e/5 + 3*b **2*c*d**3/5) + x**4*(a**3*e**3/4 + 9*a**2*b*d*e**2/4 + 9*a**2*c*d**2*e/4 + 9*a*b**2*d**2*e/4 + 3*a*b*c*d**3/2 + b**3*d**3/4) + x**3*(a**3*d*e**2 + 3*a**2*b*d**2*e + a**2*c*d**3 + a*b**2*d**3) + x**2*(3*a**3*d**2*e/2 + 3*a **2*b*d**3/2)
Time = 0.04 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.35 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, {\left (c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{9} + \frac {3}{8} \, {\left (c^{3} d^{2} e + 3 \, b c^{2} d e^{2} + {\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, b c^{2} d^{2} e + 9 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} + {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x^{7} + a^{3} d^{3} x + \frac {1}{2} \, {\left (b c^{2} d^{3} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e + {\left (b^{3} + 6 \, a b c\right )} d e^{2} + {\left (a b^{2} + a^{2} c\right )} e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (a^{2} b e^{3} + {\left (b^{2} c + a c^{2}\right )} d^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e + 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (9 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{3} + 9 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e\right )} x^{4} + {\left (3 \, a^{2} b d^{2} e + a^{3} d e^{2} + {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b d^{3} + a^{3} d^{2} e\right )} x^{2} \] Input:
integrate((e*x+d)^3*(c*x^2+b*x+a)^3,x, algorithm="maxima")
Output:
1/10*c^3*e^3*x^10 + 1/3*(c^3*d*e^2 + b*c^2*e^3)*x^9 + 3/8*(c^3*d^2*e + 3*b *c^2*d*e^2 + (b^2*c + a*c^2)*e^3)*x^8 + 1/7*(c^3*d^3 + 9*b*c^2*d^2*e + 9*( b^2*c + a*c^2)*d*e^2 + (b^3 + 6*a*b*c)*e^3)*x^7 + a^3*d^3*x + 1/2*(b*c^2*d ^3 + 3*(b^2*c + a*c^2)*d^2*e + (b^3 + 6*a*b*c)*d*e^2 + (a*b^2 + a^2*c)*e^3 )*x^6 + 3/5*(a^2*b*e^3 + (b^2*c + a*c^2)*d^3 + (b^3 + 6*a*b*c)*d^2*e + 3*( a*b^2 + a^2*c)*d*e^2)*x^5 + 1/4*(9*a^2*b*d*e^2 + a^3*e^3 + (b^3 + 6*a*b*c) *d^3 + 9*(a*b^2 + a^2*c)*d^2*e)*x^4 + (3*a^2*b*d^2*e + a^3*d*e^2 + (a*b^2 + a^2*c)*d^3)*x^3 + 3/2*(a^2*b*d^3 + a^3*d^2*e)*x^2
Time = 0.32 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.76 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, c^{3} d e^{2} x^{9} + \frac {1}{3} \, b c^{2} e^{3} x^{9} + \frac {3}{8} \, c^{3} d^{2} e x^{8} + \frac {9}{8} \, b c^{2} d e^{2} x^{8} + \frac {3}{8} \, b^{2} c e^{3} x^{8} + \frac {3}{8} \, a c^{2} e^{3} x^{8} + \frac {1}{7} \, c^{3} d^{3} x^{7} + \frac {9}{7} \, b c^{2} d^{2} e x^{7} + \frac {9}{7} \, b^{2} c d e^{2} x^{7} + \frac {9}{7} \, a c^{2} d e^{2} x^{7} + \frac {1}{7} \, b^{3} e^{3} x^{7} + \frac {6}{7} \, a b c e^{3} x^{7} + \frac {1}{2} \, b c^{2} d^{3} x^{6} + \frac {3}{2} \, b^{2} c d^{2} e x^{6} + \frac {3}{2} \, a c^{2} d^{2} e x^{6} + \frac {1}{2} \, b^{3} d e^{2} x^{6} + 3 \, a b c d e^{2} x^{6} + \frac {1}{2} \, a b^{2} e^{3} x^{6} + \frac {1}{2} \, a^{2} c e^{3} x^{6} + \frac {3}{5} \, b^{2} c d^{3} x^{5} + \frac {3}{5} \, a c^{2} d^{3} x^{5} + \frac {3}{5} \, b^{3} d^{2} e x^{5} + \frac {18}{5} \, a b c d^{2} e x^{5} + \frac {9}{5} \, a b^{2} d e^{2} x^{5} + \frac {9}{5} \, a^{2} c d e^{2} x^{5} + \frac {3}{5} \, a^{2} b e^{3} x^{5} + \frac {1}{4} \, b^{3} d^{3} x^{4} + \frac {3}{2} \, a b c d^{3} x^{4} + \frac {9}{4} \, a b^{2} d^{2} e x^{4} + \frac {9}{4} \, a^{2} c d^{2} e x^{4} + \frac {9}{4} \, a^{2} b d e^{2} x^{4} + \frac {1}{4} \, a^{3} e^{3} x^{4} + a b^{2} d^{3} x^{3} + a^{2} c d^{3} x^{3} + 3 \, a^{2} b d^{2} e x^{3} + a^{3} d e^{2} x^{3} + \frac {3}{2} \, a^{2} b d^{3} x^{2} + \frac {3}{2} \, a^{3} d^{2} e x^{2} + a^{3} d^{3} x \] Input:
integrate((e*x+d)^3*(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
1/10*c^3*e^3*x^10 + 1/3*c^3*d*e^2*x^9 + 1/3*b*c^2*e^3*x^9 + 3/8*c^3*d^2*e* x^8 + 9/8*b*c^2*d*e^2*x^8 + 3/8*b^2*c*e^3*x^8 + 3/8*a*c^2*e^3*x^8 + 1/7*c^ 3*d^3*x^7 + 9/7*b*c^2*d^2*e*x^7 + 9/7*b^2*c*d*e^2*x^7 + 9/7*a*c^2*d*e^2*x^ 7 + 1/7*b^3*e^3*x^7 + 6/7*a*b*c*e^3*x^7 + 1/2*b*c^2*d^3*x^6 + 3/2*b^2*c*d^ 2*e*x^6 + 3/2*a*c^2*d^2*e*x^6 + 1/2*b^3*d*e^2*x^6 + 3*a*b*c*d*e^2*x^6 + 1/ 2*a*b^2*e^3*x^6 + 1/2*a^2*c*e^3*x^6 + 3/5*b^2*c*d^3*x^5 + 3/5*a*c^2*d^3*x^ 5 + 3/5*b^3*d^2*e*x^5 + 18/5*a*b*c*d^2*e*x^5 + 9/5*a*b^2*d*e^2*x^5 + 9/5*a ^2*c*d*e^2*x^5 + 3/5*a^2*b*e^3*x^5 + 1/4*b^3*d^3*x^4 + 3/2*a*b*c*d^3*x^4 + 9/4*a*b^2*d^2*e*x^4 + 9/4*a^2*c*d^2*e*x^4 + 9/4*a^2*b*d*e^2*x^4 + 1/4*a^3 *e^3*x^4 + a*b^2*d^3*x^3 + a^2*c*d^3*x^3 + 3*a^2*b*d^2*e*x^3 + a^3*d*e^2*x ^3 + 3/2*a^2*b*d^3*x^2 + 3/2*a^3*d^2*e*x^2 + a^3*d^3*x
Time = 0.08 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.40 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=x^4\,\left (\frac {a^3\,e^3}{4}+\frac {9\,a^2\,b\,d\,e^2}{4}+\frac {9\,c\,a^2\,d^2\,e}{4}+\frac {9\,a\,b^2\,d^2\,e}{4}+\frac {3\,c\,a\,b\,d^3}{2}+\frac {b^3\,d^3}{4}\right )+x^7\,\left (\frac {b^3\,e^3}{7}+\frac {9\,b^2\,c\,d\,e^2}{7}+\frac {9\,b\,c^2\,d^2\,e}{7}+\frac {6\,a\,b\,c\,e^3}{7}+\frac {c^3\,d^3}{7}+\frac {9\,a\,c^2\,d\,e^2}{7}\right )+x^5\,\left (\frac {3\,a^2\,b\,e^3}{5}+\frac {9\,a^2\,c\,d\,e^2}{5}+\frac {9\,a\,b^2\,d\,e^2}{5}+\frac {18\,a\,b\,c\,d^2\,e}{5}+\frac {3\,a\,c^2\,d^3}{5}+\frac {3\,b^3\,d^2\,e}{5}+\frac {3\,b^2\,c\,d^3}{5}\right )+x^6\,\left (\frac {a^2\,c\,e^3}{2}+\frac {a\,b^2\,e^3}{2}+3\,a\,b\,c\,d\,e^2+\frac {3\,a\,c^2\,d^2\,e}{2}+\frac {b^3\,d\,e^2}{2}+\frac {3\,b^2\,c\,d^2\,e}{2}+\frac {b\,c^2\,d^3}{2}\right )+a^3\,d^3\,x+\frac {c^3\,e^3\,x^{10}}{10}+\frac {3\,a^2\,d^2\,x^2\,\left (a\,e+b\,d\right )}{2}+\frac {c^2\,e^2\,x^9\,\left (b\,e+c\,d\right )}{3}+a\,d\,x^3\,\left (a^2\,e^2+3\,a\,b\,d\,e+c\,a\,d^2+b^2\,d^2\right )+\frac {3\,c\,e\,x^8\,\left (b^2\,e^2+3\,b\,c\,d\,e+c^2\,d^2+a\,c\,e^2\right )}{8} \] Input:
int((d + e*x)^3*(a + b*x + c*x^2)^3,x)
Output:
x^4*((a^3*e^3)/4 + (b^3*d^3)/4 + (3*a*b*c*d^3)/2 + (9*a*b^2*d^2*e)/4 + (9* a^2*b*d*e^2)/4 + (9*a^2*c*d^2*e)/4) + x^7*((b^3*e^3)/7 + (c^3*d^3)/7 + (6* a*b*c*e^3)/7 + (9*a*c^2*d*e^2)/7 + (9*b*c^2*d^2*e)/7 + (9*b^2*c*d*e^2)/7) + x^5*((3*a*c^2*d^3)/5 + (3*a^2*b*e^3)/5 + (3*b^2*c*d^3)/5 + (3*b^3*d^2*e) /5 + (9*a*b^2*d*e^2)/5 + (9*a^2*c*d*e^2)/5 + (18*a*b*c*d^2*e)/5) + x^6*((a *b^2*e^3)/2 + (b*c^2*d^3)/2 + (a^2*c*e^3)/2 + (b^3*d*e^2)/2 + (3*a*c^2*d^2 *e)/2 + (3*b^2*c*d^2*e)/2 + 3*a*b*c*d*e^2) + a^3*d^3*x + (c^3*e^3*x^10)/10 + (3*a^2*d^2*x^2*(a*e + b*d))/2 + (c^2*e^2*x^9*(b*e + c*d))/3 + a*d*x^3*( a^2*e^2 + b^2*d^2 + a*c*d^2 + 3*a*b*d*e) + (3*c*e*x^8*(b^2*e^2 + c^2*d^2 + a*c*e^2 + 3*b*c*d*e))/8
Time = 0.21 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.77 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^3 \, dx=\frac {x \left (84 c^{3} e^{3} x^{9}+280 b \,c^{2} e^{3} x^{8}+280 c^{3} d \,e^{2} x^{8}+315 a \,c^{2} e^{3} x^{7}+315 b^{2} c \,e^{3} x^{7}+945 b \,c^{2} d \,e^{2} x^{7}+315 c^{3} d^{2} e \,x^{7}+720 a b c \,e^{3} x^{6}+1080 a \,c^{2} d \,e^{2} x^{6}+120 b^{3} e^{3} x^{6}+1080 b^{2} c d \,e^{2} x^{6}+1080 b \,c^{2} d^{2} e \,x^{6}+120 c^{3} d^{3} x^{6}+420 a^{2} c \,e^{3} x^{5}+420 a \,b^{2} e^{3} x^{5}+2520 a b c d \,e^{2} x^{5}+1260 a \,c^{2} d^{2} e \,x^{5}+420 b^{3} d \,e^{2} x^{5}+1260 b^{2} c \,d^{2} e \,x^{5}+420 b \,c^{2} d^{3} x^{5}+504 a^{2} b \,e^{3} x^{4}+1512 a^{2} c d \,e^{2} x^{4}+1512 a \,b^{2} d \,e^{2} x^{4}+3024 a b c \,d^{2} e \,x^{4}+504 a \,c^{2} d^{3} x^{4}+504 b^{3} d^{2} e \,x^{4}+504 b^{2} c \,d^{3} x^{4}+210 a^{3} e^{3} x^{3}+1890 a^{2} b d \,e^{2} x^{3}+1890 a^{2} c \,d^{2} e \,x^{3}+1890 a \,b^{2} d^{2} e \,x^{3}+1260 a b c \,d^{3} x^{3}+210 b^{3} d^{3} x^{3}+840 a^{3} d \,e^{2} x^{2}+2520 a^{2} b \,d^{2} e \,x^{2}+840 a^{2} c \,d^{3} x^{2}+840 a \,b^{2} d^{3} x^{2}+1260 a^{3} d^{2} e x +1260 a^{2} b \,d^{3} x +840 a^{3} d^{3}\right )}{840} \] Input:
int((e*x+d)^3*(c*x^2+b*x+a)^3,x)
Output:
(x*(840*a**3*d**3 + 1260*a**3*d**2*e*x + 840*a**3*d*e**2*x**2 + 210*a**3*e **3*x**3 + 1260*a**2*b*d**3*x + 2520*a**2*b*d**2*e*x**2 + 1890*a**2*b*d*e* *2*x**3 + 504*a**2*b*e**3*x**4 + 840*a**2*c*d**3*x**2 + 1890*a**2*c*d**2*e *x**3 + 1512*a**2*c*d*e**2*x**4 + 420*a**2*c*e**3*x**5 + 840*a*b**2*d**3*x **2 + 1890*a*b**2*d**2*e*x**3 + 1512*a*b**2*d*e**2*x**4 + 420*a*b**2*e**3* x**5 + 1260*a*b*c*d**3*x**3 + 3024*a*b*c*d**2*e*x**4 + 2520*a*b*c*d*e**2*x **5 + 720*a*b*c*e**3*x**6 + 504*a*c**2*d**3*x**4 + 1260*a*c**2*d**2*e*x**5 + 1080*a*c**2*d*e**2*x**6 + 315*a*c**2*e**3*x**7 + 210*b**3*d**3*x**3 + 5 04*b**3*d**2*e*x**4 + 420*b**3*d*e**2*x**5 + 120*b**3*e**3*x**6 + 504*b**2 *c*d**3*x**4 + 1260*b**2*c*d**2*e*x**5 + 1080*b**2*c*d*e**2*x**6 + 315*b** 2*c*e**3*x**7 + 420*b*c**2*d**3*x**5 + 1080*b*c**2*d**2*e*x**6 + 945*b*c** 2*d*e**2*x**7 + 280*b*c**2*e**3*x**8 + 120*c**3*d**3*x**6 + 315*c**3*d**2* e*x**7 + 280*c**3*d*e**2*x**8 + 84*c**3*e**3*x**9))/840