Integrand size = 24, antiderivative size = 251 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=a^3 b d x+\frac {1}{2} a^2 \left (3 b^2 d+2 a c d+a b e\right ) x^2+\frac {1}{3} a \left (3 b^3 d+9 a b c d+3 a b^2 e+2 a^2 c e\right ) x^3+\frac {1}{4} \left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+3 a b^3 e+9 a^2 b c e\right ) x^4+\frac {1}{5} \left (5 b^3 c d+15 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^5+\frac {1}{6} c \left (9 b^2 c d+6 a c^2 d+5 b^3 e+15 a b c e\right ) x^6+\frac {1}{7} c^2 \left (7 b c d+9 b^2 e+6 a c e\right ) x^7+\frac {1}{8} c^3 (2 c d+7 b e) x^8+\frac {2}{9} c^4 e x^9 \]
a^3*b*d*x+1/2*a^2*(a*b*e+2*a*c*d+3*b^2*d)*x^2+1/3*a*(2*a^2*c*e+3*a*b^2*e+9 *a*b*c*d+3*b^3*d)*x^3+1/4*(9*a^2*b*c*e+6*a^2*c^2*d+3*a*b^3*e+12*a*b^2*c*d+ b^4*d)*x^4+1/5*(6*a^2*c^2*e+12*a*b^2*c*e+15*a*b*c^2*d+b^4*e+5*b^3*c*d)*x^5 +1/6*c*(15*a*b*c*e+6*a*c^2*d+5*b^3*e+9*b^2*c*d)*x^6+1/7*c^2*(6*a*c*e+9*b^2 *e+7*b*c*d)*x^7+1/8*c^3*(7*b*e+2*c*d)*x^8+2/9*c^4*e*x^9
Time = 0.06 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=a^3 b d x+\frac {1}{2} a^2 \left (3 b^2 d+2 a c d+a b e\right ) x^2+\frac {1}{3} a \left (3 b^3 d+9 a b c d+3 a b^2 e+2 a^2 c e\right ) x^3+\frac {1}{4} \left (b^4 d+12 a b^2 c d+6 a^2 c^2 d+3 a b^3 e+9 a^2 b c e\right ) x^4+\frac {1}{5} \left (5 b^3 c d+15 a b c^2 d+b^4 e+12 a b^2 c e+6 a^2 c^2 e\right ) x^5+\frac {1}{6} c \left (9 b^2 c d+6 a c^2 d+5 b^3 e+15 a b c e\right ) x^6+\frac {1}{7} c^2 \left (7 b c d+9 b^2 e+6 a c e\right ) x^7+\frac {1}{8} c^3 (2 c d+7 b e) x^8+\frac {2}{9} c^4 e x^9 \]
a^3*b*d*x + (a^2*(3*b^2*d + 2*a*c*d + a*b*e)*x^2)/2 + (a*(3*b^3*d + 9*a*b* c*d + 3*a*b^2*e + 2*a^2*c*e)*x^3)/3 + ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 3*a*b^3*e + 9*a^2*b*c*e)*x^4)/4 + ((5*b^3*c*d + 15*a*b*c^2*d + b^4*e + 12*a*b^2*c*e + 6*a^2*c^2*e)*x^5)/5 + (c*(9*b^2*c*d + 6*a*c^2*d + 5*b^3*e + 15*a*b*c*e)*x^6)/6 + (c^2*(7*b*c*d + 9*b^2*e + 6*a*c*e)*x^7)/7 + (c^3*(2* c*d + 7*b*e)*x^8)/8 + (2*c^4*e*x^9)/9
Time = 0.51 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1195, 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.
| \(\displaystyle \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (a^3 b d+a^2 x \left (a b e+2 a c d+3 b^2 d\right )+a x^2 \left (2 a^2 c e+3 a b^2 e+9 a b c d+3 b^3 d\right )+x^4 \left (6 a^2 c^2 e+12 a b^2 c e+15 a b c^2 d+b^4 e+5 b^3 c d\right )+x^3 \left (9 a^2 b c e+6 a^2 c^2 d+3 a b^3 e+12 a b^2 c d+b^4 d\right )+c^2 x^6 \left (6 a c e+9 b^2 e+7 b c d\right )+c x^5 \left (15 a b c e+6 a c^2 d+5 b^3 e+9 b^2 c d\right )+c^3 x^7 (7 b e+2 c d)+2 c^4 e x^8\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^3 b d x+\frac {1}{2} a^2 x^2 \left (a b e+2 a c d+3 b^2 d\right )+\frac {1}{3} a x^3 \left (2 a^2 c e+3 a b^2 e+9 a b c d+3 b^3 d\right )+\frac {1}{5} x^5 \left (6 a^2 c^2 e+12 a b^2 c e+15 a b c^2 d+b^4 e+5 b^3 c d\right )+\frac {1}{4} x^4 \left (9 a^2 b c e+6 a^2 c^2 d+3 a b^3 e+12 a b^2 c d+b^4 d\right )+\frac {1}{7} c^2 x^7 \left (6 a c e+9 b^2 e+7 b c d\right )+\frac {1}{6} c x^6 \left (15 a b c e+6 a c^2 d+5 b^3 e+9 b^2 c d\right )+\frac {1}{8} c^3 x^8 (7 b e+2 c d)+\frac {2}{9} c^4 e x^9\) |
a^3*b*d*x + (a^2*(3*b^2*d + 2*a*c*d + a*b*e)*x^2)/2 + (a*(3*b^3*d + 9*a*b* c*d + 3*a*b^2*e + 2*a^2*c*e)*x^3)/3 + ((b^4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 3*a*b^3*e + 9*a^2*b*c*e)*x^4)/4 + ((5*b^3*c*d + 15*a*b*c^2*d + b^4*e + 12*a*b^2*c*e + 6*a^2*c^2*e)*x^5)/5 + (c*(9*b^2*c*d + 6*a*c^2*d + 5*b^3*e + 15*a*b*c*e)*x^6)/6 + (c^2*(7*b*c*d + 9*b^2*e + 6*a*c*e)*x^7)/7 + (c^3*(2* c*d + 7*b*e)*x^8)/8 + (2*c^4*e*x^9)/9
3.16.17.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.38 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.97
| method | result | size |
| norman | \(\frac {2 c^{4} e \,x^{9}}{9}+\left (\frac {7}{8} c^{3} e b +\frac {1}{4} c^{4} d \right ) x^{8}+\left (\frac {6}{7} a \,c^{3} e +\frac {9}{7} b^{2} c^{2} e +b d \,c^{3}\right ) x^{7}+\left (\frac {5}{2} a b \,c^{2} e +a \,c^{3} d +\frac {5}{6} b^{3} c e +\frac {3}{2} b^{2} d \,c^{2}\right ) x^{6}+\left (\frac {6}{5} a^{2} c^{2} e +\frac {12}{5} a \,b^{2} c e +3 a b \,c^{2} d +\frac {1}{5} b^{4} e +b^{3} c d \right ) x^{5}+\left (\frac {9}{4} a^{2} b c e +\frac {3}{2} a^{2} c^{2} d +\frac {3}{4} a \,b^{3} e +3 a \,b^{2} c d +\frac {1}{4} b^{4} d \right ) x^{4}+\left (\frac {2}{3} c e \,a^{3}+a^{2} b^{2} e +3 a^{2} b c d +a \,b^{3} d \right ) x^{3}+\left (\frac {1}{2} a^{3} b e +a^{3} c d +\frac {3}{2} b^{2} d \,a^{2}\right ) x^{2}+a^{3} b d x\) | \(244\) |
| gosper | \(\frac {2}{9} c^{4} e \,x^{9}+\frac {1}{4} x^{8} c^{4} d +\frac {1}{5} x^{5} b^{4} e +\frac {1}{4} x^{4} b^{4} d +a^{3} b d x +\frac {3}{2} x^{2} b^{2} d \,a^{2}+x^{2} a^{3} c d +\frac {1}{2} x^{2} a^{3} b e +x^{3} a \,b^{3} d +x^{3} a^{2} b^{2} e +\frac {2}{3} x^{3} c e \,a^{3}+\frac {3}{4} x^{4} a \,b^{3} e +\frac {3}{2} x^{4} a^{2} c^{2} d +\frac {6}{5} x^{5} a^{2} c^{2} e +x^{5} b^{3} c d +\frac {5}{6} x^{6} b^{3} c e +\frac {3}{2} x^{6} b^{2} d \,c^{2}+x^{6} a \,c^{3} d +\frac {9}{7} x^{7} b^{2} c^{2} e +x^{7} b d \,c^{3}+\frac {6}{7} x^{7} a \,c^{3} e +\frac {7}{8} x^{8} c^{3} e b +\frac {12}{5} x^{5} a \,b^{2} c e +3 x^{5} a b \,c^{2} d +\frac {9}{4} x^{4} a^{2} b c e +3 x^{4} a \,b^{2} c d +3 x^{3} a^{2} b c d +\frac {5}{2} x^{6} a b \,c^{2} e\) | \(287\) |
| risch | \(\frac {2}{9} c^{4} e \,x^{9}+\frac {1}{4} x^{8} c^{4} d +\frac {1}{5} x^{5} b^{4} e +\frac {1}{4} x^{4} b^{4} d +a^{3} b d x +\frac {3}{2} x^{2} b^{2} d \,a^{2}+x^{2} a^{3} c d +\frac {1}{2} x^{2} a^{3} b e +x^{3} a \,b^{3} d +x^{3} a^{2} b^{2} e +\frac {2}{3} x^{3} c e \,a^{3}+\frac {3}{4} x^{4} a \,b^{3} e +\frac {3}{2} x^{4} a^{2} c^{2} d +\frac {6}{5} x^{5} a^{2} c^{2} e +x^{5} b^{3} c d +\frac {5}{6} x^{6} b^{3} c e +\frac {3}{2} x^{6} b^{2} d \,c^{2}+x^{6} a \,c^{3} d +\frac {9}{7} x^{7} b^{2} c^{2} e +x^{7} b d \,c^{3}+\frac {6}{7} x^{7} a \,c^{3} e +\frac {7}{8} x^{8} c^{3} e b +\frac {12}{5} x^{5} a \,b^{2} c e +3 x^{5} a b \,c^{2} d +\frac {9}{4} x^{4} a^{2} b c e +3 x^{4} a \,b^{2} c d +3 x^{3} a^{2} b c d +\frac {5}{2} x^{6} a b \,c^{2} e\) | \(287\) |
| parallelrisch | \(\frac {2}{9} c^{4} e \,x^{9}+\frac {1}{4} x^{8} c^{4} d +\frac {1}{5} x^{5} b^{4} e +\frac {1}{4} x^{4} b^{4} d +a^{3} b d x +\frac {3}{2} x^{2} b^{2} d \,a^{2}+x^{2} a^{3} c d +\frac {1}{2} x^{2} a^{3} b e +x^{3} a \,b^{3} d +x^{3} a^{2} b^{2} e +\frac {2}{3} x^{3} c e \,a^{3}+\frac {3}{4} x^{4} a \,b^{3} e +\frac {3}{2} x^{4} a^{2} c^{2} d +\frac {6}{5} x^{5} a^{2} c^{2} e +x^{5} b^{3} c d +\frac {5}{6} x^{6} b^{3} c e +\frac {3}{2} x^{6} b^{2} d \,c^{2}+x^{6} a \,c^{3} d +\frac {9}{7} x^{7} b^{2} c^{2} e +x^{7} b d \,c^{3}+\frac {6}{7} x^{7} a \,c^{3} e +\frac {7}{8} x^{8} c^{3} e b +\frac {12}{5} x^{5} a \,b^{2} c e +3 x^{5} a b \,c^{2} d +\frac {9}{4} x^{4} a^{2} b c e +3 x^{4} a \,b^{2} c d +3 x^{3} a^{2} b c d +\frac {5}{2} x^{6} a b \,c^{2} e\) | \(287\) |
| default | \(\frac {2 c^{4} e \,x^{9}}{9}+\frac {\left (\left (b e +2 c d \right ) c^{3}+6 c^{3} e b \right ) x^{8}}{8}+\frac {\left (b d \,c^{3}+3 \left (b e +2 c d \right ) b \,c^{2}+2 c e \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (3 b^{2} d \,c^{2}+\left (b e +2 c d \right ) \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+2 c e \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (b d \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+\left (b e +2 c d \right ) \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+2 c e \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )\right ) x^{5}}{5}+\frac {\left (b d \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+\left (b e +2 c d \right ) \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )+6 a^{2} b c e \right ) x^{4}}{4}+\frac {\left (b d \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )+3 \left (b e +2 c d \right ) b \,a^{2}+2 c e \,a^{3}\right ) x^{3}}{3}+\frac {\left (3 b^{2} d \,a^{2}+\left (b e +2 c d \right ) a^{3}\right ) x^{2}}{2}+a^{3} b d x\) | \(386\) |
2/9*c^4*e*x^9+(7/8*c^3*e*b+1/4*c^4*d)*x^8+(6/7*a*c^3*e+9/7*b^2*c^2*e+b*d*c ^3)*x^7+(5/2*a*b*c^2*e+a*c^3*d+5/6*b^3*c*e+3/2*b^2*d*c^2)*x^6+(6/5*a^2*c^2 *e+12/5*a*b^2*c*e+3*a*b*c^2*d+1/5*b^4*e+b^3*c*d)*x^5+(9/4*a^2*b*c*e+3/2*a^ 2*c^2*d+3/4*a*b^3*e+3*a*b^2*c*d+1/4*b^4*d)*x^4+(2/3*c*e*a^3+a^2*b^2*e+3*a^ 2*b*c*d+a*b^3*d)*x^3+(1/2*a^3*b*e+a^3*c*d+3/2*b^2*d*a^2)*x^2+a^3*b*d*x
Time = 0.28 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.04 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=\frac {2}{9} \, c^{4} e x^{9} + \frac {1}{8} \, {\left (2 \, c^{4} d + 7 \, b c^{3} e\right )} x^{8} + \frac {1}{7} \, {\left (7 \, b c^{3} d + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e\right )} x^{6} + a^{3} b d x + \frac {1}{5} \, {\left (5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e\right )} x^{5} + \frac {1}{4} \, {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (a^{3} b e + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} x^{2} \]
2/9*c^4*e*x^9 + 1/8*(2*c^4*d + 7*b*c^3*e)*x^8 + 1/7*(7*b*c^3*d + 3*(3*b^2* c^2 + 2*a*c^3)*e)*x^7 + 1/6*(3*(3*b^2*c^2 + 2*a*c^3)*d + 5*(b^3*c + 3*a*b* c^2)*e)*x^6 + a^3*b*d*x + 1/5*(5*(b^3*c + 3*a*b*c^2)*d + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e)*x^5 + 1/4*((b^4 + 12*a*b^2*c + 6*a^2*c^2)*d + 3*(a*b^3 + 3*a^2*b*c)*e)*x^4 + 1/3*(3*(a*b^3 + 3*a^2*b*c)*d + (3*a^2*b^2 + 2*a^3*c)*e )*x^3 + 1/2*(a^3*b*e + (3*a^2*b^2 + 2*a^3*c)*d)*x^2
Time = 0.03 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.16 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=a^{3} b d x + \frac {2 c^{4} e x^{9}}{9} + x^{8} \cdot \left (\frac {7 b c^{3} e}{8} + \frac {c^{4} d}{4}\right ) + x^{7} \cdot \left (\frac {6 a c^{3} e}{7} + \frac {9 b^{2} c^{2} e}{7} + b c^{3} d\right ) + x^{6} \cdot \left (\frac {5 a b c^{2} e}{2} + a c^{3} d + \frac {5 b^{3} c e}{6} + \frac {3 b^{2} c^{2} d}{2}\right ) + x^{5} \cdot \left (\frac {6 a^{2} c^{2} e}{5} + \frac {12 a b^{2} c e}{5} + 3 a b c^{2} d + \frac {b^{4} e}{5} + b^{3} c d\right ) + x^{4} \cdot \left (\frac {9 a^{2} b c e}{4} + \frac {3 a^{2} c^{2} d}{2} + \frac {3 a b^{3} e}{4} + 3 a b^{2} c d + \frac {b^{4} d}{4}\right ) + x^{3} \cdot \left (\frac {2 a^{3} c e}{3} + a^{2} b^{2} e + 3 a^{2} b c d + a b^{3} d\right ) + x^{2} \left (\frac {a^{3} b e}{2} + a^{3} c d + \frac {3 a^{2} b^{2} d}{2}\right ) \]
a**3*b*d*x + 2*c**4*e*x**9/9 + x**8*(7*b*c**3*e/8 + c**4*d/4) + x**7*(6*a* c**3*e/7 + 9*b**2*c**2*e/7 + b*c**3*d) + x**6*(5*a*b*c**2*e/2 + a*c**3*d + 5*b**3*c*e/6 + 3*b**2*c**2*d/2) + x**5*(6*a**2*c**2*e/5 + 12*a*b**2*c*e/5 + 3*a*b*c**2*d + b**4*e/5 + b**3*c*d) + x**4*(9*a**2*b*c*e/4 + 3*a**2*c** 2*d/2 + 3*a*b**3*e/4 + 3*a*b**2*c*d + b**4*d/4) + x**3*(2*a**3*c*e/3 + a** 2*b**2*e + 3*a**2*b*c*d + a*b**3*d) + x**2*(a**3*b*e/2 + a**3*c*d + 3*a**2 *b**2*d/2)
Time = 0.19 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.04 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=\frac {2}{9} \, c^{4} e x^{9} + \frac {1}{8} \, {\left (2 \, c^{4} d + 7 \, b c^{3} e\right )} x^{8} + \frac {1}{7} \, {\left (7 \, b c^{3} d + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e\right )} x^{6} + a^{3} b d x + \frac {1}{5} \, {\left (5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e\right )} x^{5} + \frac {1}{4} \, {\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (a^{3} b e + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} x^{2} \]
2/9*c^4*e*x^9 + 1/8*(2*c^4*d + 7*b*c^3*e)*x^8 + 1/7*(7*b*c^3*d + 3*(3*b^2* c^2 + 2*a*c^3)*e)*x^7 + 1/6*(3*(3*b^2*c^2 + 2*a*c^3)*d + 5*(b^3*c + 3*a*b* c^2)*e)*x^6 + a^3*b*d*x + 1/5*(5*(b^3*c + 3*a*b*c^2)*d + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e)*x^5 + 1/4*((b^4 + 12*a*b^2*c + 6*a^2*c^2)*d + 3*(a*b^3 + 3*a^2*b*c)*e)*x^4 + 1/3*(3*(a*b^3 + 3*a^2*b*c)*d + (3*a^2*b^2 + 2*a^3*c)*e )*x^3 + 1/2*(a^3*b*e + (3*a^2*b^2 + 2*a^3*c)*d)*x^2
Time = 0.26 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.14 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=\frac {2}{9} \, c^{4} e x^{9} + \frac {1}{4} \, c^{4} d x^{8} + \frac {7}{8} \, b c^{3} e x^{8} + b c^{3} d x^{7} + \frac {9}{7} \, b^{2} c^{2} e x^{7} + \frac {6}{7} \, a c^{3} e x^{7} + \frac {3}{2} \, b^{2} c^{2} d x^{6} + a c^{3} d x^{6} + \frac {5}{6} \, b^{3} c e x^{6} + \frac {5}{2} \, a b c^{2} e x^{6} + b^{3} c d x^{5} + 3 \, a b c^{2} d x^{5} + \frac {1}{5} \, b^{4} e x^{5} + \frac {12}{5} \, a b^{2} c e x^{5} + \frac {6}{5} \, a^{2} c^{2} e x^{5} + \frac {1}{4} \, b^{4} d x^{4} + 3 \, a b^{2} c d x^{4} + \frac {3}{2} \, a^{2} c^{2} d x^{4} + \frac {3}{4} \, a b^{3} e x^{4} + \frac {9}{4} \, a^{2} b c e x^{4} + a b^{3} d x^{3} + 3 \, a^{2} b c d x^{3} + a^{2} b^{2} e x^{3} + \frac {2}{3} \, a^{3} c e x^{3} + \frac {3}{2} \, a^{2} b^{2} d x^{2} + a^{3} c d x^{2} + \frac {1}{2} \, a^{3} b e x^{2} + a^{3} b d x \]
2/9*c^4*e*x^9 + 1/4*c^4*d*x^8 + 7/8*b*c^3*e*x^8 + b*c^3*d*x^7 + 9/7*b^2*c^ 2*e*x^7 + 6/7*a*c^3*e*x^7 + 3/2*b^2*c^2*d*x^6 + a*c^3*d*x^6 + 5/6*b^3*c*e* x^6 + 5/2*a*b*c^2*e*x^6 + b^3*c*d*x^5 + 3*a*b*c^2*d*x^5 + 1/5*b^4*e*x^5 + 12/5*a*b^2*c*e*x^5 + 6/5*a^2*c^2*e*x^5 + 1/4*b^4*d*x^4 + 3*a*b^2*c*d*x^4 + 3/2*a^2*c^2*d*x^4 + 3/4*a*b^3*e*x^4 + 9/4*a^2*b*c*e*x^4 + a*b^3*d*x^3 + 3 *a^2*b*c*d*x^3 + a^2*b^2*e*x^3 + 2/3*a^3*c*e*x^3 + 3/2*a^2*b^2*d*x^2 + a^3 *c*d*x^2 + 1/2*a^3*b*e*x^2 + a^3*b*d*x
Time = 10.71 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.97 \[ \int (b+2 c x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=x^8\,\left (\frac {d\,c^4}{4}+\frac {7\,b\,e\,c^3}{8}\right )+x^2\,\left (\frac {e\,a^3\,b}{2}+c\,d\,a^3+\frac {3\,d\,a^2\,b^2}{2}\right )+x^7\,\left (\frac {9\,e\,b^2\,c^2}{7}+d\,b\,c^3+\frac {6\,a\,e\,c^3}{7}\right )+x^4\,\left (\frac {9\,e\,a^2\,b\,c}{4}+\frac {3\,d\,a^2\,c^2}{2}+\frac {3\,e\,a\,b^3}{4}+3\,d\,a\,b^2\,c+\frac {d\,b^4}{4}\right )+x^5\,\left (\frac {6\,e\,a^2\,c^2}{5}+\frac {12\,e\,a\,b^2\,c}{5}+3\,d\,a\,b\,c^2+\frac {e\,b^4}{5}+d\,b^3\,c\right )+x^3\,\left (\frac {2\,c\,e\,a^3}{3}+e\,a^2\,b^2+3\,c\,d\,a^2\,b+d\,a\,b^3\right )+x^6\,\left (\frac {5\,e\,b^3\,c}{6}+\frac {3\,d\,b^2\,c^2}{2}+\frac {5\,a\,e\,b\,c^2}{2}+a\,d\,c^3\right )+\frac {2\,c^4\,e\,x^9}{9}+a^3\,b\,d\,x \]
x^8*((c^4*d)/4 + (7*b*c^3*e)/8) + x^2*((3*a^2*b^2*d)/2 + (a^3*b*e)/2 + a^3 *c*d) + x^7*((9*b^2*c^2*e)/7 + (6*a*c^3*e)/7 + b*c^3*d) + x^4*((b^4*d)/4 + (3*a^2*c^2*d)/2 + (3*a*b^3*e)/4 + 3*a*b^2*c*d + (9*a^2*b*c*e)/4) + x^5*(( b^4*e)/5 + (6*a^2*c^2*e)/5 + b^3*c*d + 3*a*b*c^2*d + (12*a*b^2*c*e)/5) + x ^3*(a^2*b^2*e + a*b^3*d + (2*a^3*c*e)/3 + 3*a^2*b*c*d) + x^6*((3*b^2*c^2*d )/2 + a*c^3*d + (5*b^3*c*e)/6 + (5*a*b*c^2*e)/2) + (2*c^4*e*x^9)/9 + a^3*b *d*x