Integrand size = 24, antiderivative size = 228 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} A b^2 d^3 x^3+\frac {1}{4} b d^2 (b B d+2 A c d+3 A b e) x^4+\frac {1}{5} d \left (A c^2 d^2+3 b^2 e (B d+A e)+2 b c d (B d+3 A e)\right ) x^5+\frac {1}{6} \left (A e \left (3 c^2 d^2+6 b c d e+b^2 e^2\right )+B d \left (c^2 d^2+6 b c d e+3 b^2 e^2\right )\right ) x^6+\frac {1}{7} e \left (A c e (3 c d+2 b e)+B \left (3 c^2 d^2+6 b c d e+b^2 e^2\right )\right ) x^7+\frac {1}{8} c e^2 (3 B c d+2 b B e+A c e) x^8+\frac {1}{9} B c^2 e^3 x^9 \]
1/3*A*b^2*d^3*x^3+1/4*b*d^2*(3*A*b*e+2*A*c*d+B*b*d)*x^4+1/5*d*(A*c^2*d^2+3 *b^2*e*(A*e+B*d)+2*b*c*d*(3*A*e+B*d))*x^5+1/6*(A*e*(b^2*e^2+6*b*c*d*e+3*c^ 2*d^2)+B*d*(3*b^2*e^2+6*b*c*d*e+c^2*d^2))*x^6+1/7*e*(A*c*e*(2*b*e+3*c*d)+B *(b^2*e^2+6*b*c*d*e+3*c^2*d^2))*x^7+1/8*c*e^2*(A*c*e+2*B*b*e+3*B*c*d)*x^8+ 1/9*B*c^2*e^3*x^9
Time = 0.06 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} A b^2 d^3 x^3+\frac {1}{4} b d^2 (b B d+2 A c d+3 A b e) x^4+\frac {1}{5} d \left (A c^2 d^2+3 b^2 e (B d+A e)+2 b c d (B d+3 A e)\right ) x^5+\frac {1}{6} \left (A e \left (3 c^2 d^2+6 b c d e+b^2 e^2\right )+B d \left (c^2 d^2+6 b c d e+3 b^2 e^2\right )\right ) x^6+\frac {1}{7} e \left (A c e (3 c d+2 b e)+B \left (3 c^2 d^2+6 b c d e+b^2 e^2\right )\right ) x^7+\frac {1}{8} c e^2 (3 B c d+2 b B e+A c e) x^8+\frac {1}{9} B c^2 e^3 x^9 \]
(A*b^2*d^3*x^3)/3 + (b*d^2*(b*B*d + 2*A*c*d + 3*A*b*e)*x^4)/4 + (d*(A*c^2* d^2 + 3*b^2*e*(B*d + A*e) + 2*b*c*d*(B*d + 3*A*e))*x^5)/5 + ((A*e*(3*c^2*d ^2 + 6*b*c*d*e + b^2*e^2) + B*d*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2))*x^6)/6 + (e*(A*c*e*(3*c*d + 2*b*e) + B*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2))*x^7)/7 + (c*e^2*(3*B*c*d + 2*b*B*e + A*c*e)*x^8)/8 + (B*c^2*e^3*x^9)/9
Time = 0.47 (sec) , antiderivative size = 228, 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.
| \(\displaystyle \int (A+B x) \left (b x+c x^2\right )^2 (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (e x^6 \left (A c e (2 b e+3 c d)+B \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )\right )+x^5 \left (A e \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+B d \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+d x^4 \left (3 b^2 e (A e+B d)+2 b c d (3 A e+B d)+A c^2 d^2\right )+A b^2 d^3 x^2+b d^2 x^3 (3 A b e+2 A c d+b B d)+c e^2 x^7 (A c e+2 b B e+3 B c d)+B c^2 e^3 x^8\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} e x^7 \left (A c e (2 b e+3 c d)+B \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )\right )+\frac {1}{6} x^6 \left (A e \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+B d \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac {1}{5} d x^5 \left (3 b^2 e (A e+B d)+2 b c d (3 A e+B d)+A c^2 d^2\right )+\frac {1}{3} A b^2 d^3 x^3+\frac {1}{4} b d^2 x^4 (3 A b e+2 A c d+b B d)+\frac {1}{8} c e^2 x^8 (A c e+2 b B e+3 B c d)+\frac {1}{9} B c^2 e^3 x^9\) |
(A*b^2*d^3*x^3)/3 + (b*d^2*(b*B*d + 2*A*c*d + 3*A*b*e)*x^4)/4 + (d*(A*c^2* d^2 + 3*b^2*e*(B*d + A*e) + 2*b*c*d*(B*d + 3*A*e))*x^5)/5 + ((A*e*(3*c^2*d ^2 + 6*b*c*d*e + b^2*e^2) + B*d*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2))*x^6)/6 + (e*(A*c*e*(3*c*d + 2*b*e) + B*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2))*x^7)/7 + (c*e^2*(3*B*c*d + 2*b*B*e + A*c*e)*x^8)/8 + (B*c^2*e^3*x^9)/9
3.12.15.3.1 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 = 0.20 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.08
| method | result | size |
| default | \(\frac {B \,c^{2} e^{3} x^{9}}{9}+\frac {\left (\left (A \,e^{3}+3 B d \,e^{2}\right ) c^{2}+2 B \,e^{3} b c \right ) x^{8}}{8}+\frac {\left (\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) c^{2}+2 \left (A \,e^{3}+3 B d \,e^{2}\right ) b c +B \,e^{3} b^{2}\right ) x^{7}}{7}+\frac {\left (\left (3 A \,d^{2} e +B \,d^{3}\right ) c^{2}+2 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) b c +\left (A \,e^{3}+3 B d \,e^{2}\right ) b^{2}\right ) x^{6}}{6}+\frac {\left (A \,d^{3} c^{2}+2 \left (3 A \,d^{2} e +B \,d^{3}\right ) b c +\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) b^{2}\right ) x^{5}}{5}+\frac {\left (2 A \,d^{3} b c +\left (3 A \,d^{2} e +B \,d^{3}\right ) b^{2}\right ) x^{4}}{4}+\frac {A \,b^{2} d^{3} x^{3}}{3}\) | \(247\) |
| norman | \(\frac {B \,c^{2} e^{3} x^{9}}{9}+\left (\frac {1}{8} A \,c^{2} e^{3}+\frac {1}{4} B \,e^{3} b c +\frac {3}{8} B \,c^{2} d \,e^{2}\right ) x^{8}+\left (\frac {2}{7} A b c \,e^{3}+\frac {3}{7} A \,c^{2} d \,e^{2}+\frac {1}{7} B \,e^{3} b^{2}+\frac {6}{7} B b c d \,e^{2}+\frac {3}{7} B \,c^{2} d^{2} e \right ) x^{7}+\left (\frac {1}{6} A \,b^{2} e^{3}+A b c d \,e^{2}+\frac {1}{2} A \,c^{2} d^{2} e +\frac {1}{2} B \,b^{2} d \,e^{2}+B b c \,d^{2} e +\frac {1}{6} B \,c^{2} d^{3}\right ) x^{6}+\left (\frac {3}{5} A \,b^{2} d \,e^{2}+\frac {6}{5} A b c \,d^{2} e +\frac {1}{5} A \,d^{3} c^{2}+\frac {3}{5} B \,b^{2} d^{2} e +\frac {2}{5} B b c \,d^{3}\right ) x^{5}+\left (\frac {3}{4} A \,b^{2} d^{2} e +\frac {1}{2} A \,d^{3} b c +\frac {1}{4} B \,b^{2} d^{3}\right ) x^{4}+\frac {A \,b^{2} d^{3} x^{3}}{3}\) | \(251\) |
| gosper | \(\frac {x^{3} \left (280 B \,c^{2} e^{3} x^{6}+315 x^{5} A \,c^{2} e^{3}+630 x^{5} B \,e^{3} b c +945 x^{5} B \,c^{2} d \,e^{2}+720 x^{4} A b c \,e^{3}+1080 x^{4} A \,c^{2} d \,e^{2}+360 x^{4} B \,e^{3} b^{2}+2160 x^{4} B b c d \,e^{2}+1080 x^{4} B \,c^{2} d^{2} e +420 x^{3} A \,b^{2} e^{3}+2520 x^{3} A b c d \,e^{2}+1260 x^{3} A \,c^{2} d^{2} e +1260 x^{3} B \,b^{2} d \,e^{2}+2520 x^{3} B b c \,d^{2} e +420 x^{3} B \,c^{2} d^{3}+1512 x^{2} A \,b^{2} d \,e^{2}+3024 x^{2} A b c \,d^{2} e +504 x^{2} A \,d^{3} c^{2}+1512 x^{2} B \,b^{2} d^{2} e +1008 x^{2} B b c \,d^{3}+1890 x A \,b^{2} d^{2} e +1260 x A \,d^{3} b c +630 x B \,b^{2} d^{3}+840 A \,d^{3} b^{2}\right )}{2520}\) | \(290\) |
| risch | \(\frac {1}{9} B \,c^{2} e^{3} x^{9}+\frac {1}{8} x^{8} A \,c^{2} e^{3}+\frac {1}{4} x^{8} B \,e^{3} b c +\frac {3}{8} x^{8} B \,c^{2} d \,e^{2}+\frac {2}{7} x^{7} A b c \,e^{3}+\frac {3}{7} x^{7} A \,c^{2} d \,e^{2}+\frac {1}{7} x^{7} B \,e^{3} b^{2}+\frac {6}{7} x^{7} B b c d \,e^{2}+\frac {3}{7} x^{7} B \,c^{2} d^{2} e +\frac {1}{6} x^{6} A \,b^{2} e^{3}+x^{6} A b c d \,e^{2}+\frac {1}{2} x^{6} A \,c^{2} d^{2} e +\frac {1}{2} x^{6} B \,b^{2} d \,e^{2}+x^{6} B b c \,d^{2} e +\frac {1}{6} x^{6} B \,c^{2} d^{3}+\frac {3}{5} x^{5} A \,b^{2} d \,e^{2}+\frac {6}{5} x^{5} A b c \,d^{2} e +\frac {1}{5} x^{5} A \,d^{3} c^{2}+\frac {3}{5} x^{5} B \,b^{2} d^{2} e +\frac {2}{5} x^{5} B b c \,d^{3}+\frac {3}{4} x^{4} A \,b^{2} d^{2} e +\frac {1}{2} x^{4} A \,d^{3} b c +\frac {1}{4} x^{4} B \,b^{2} d^{3}+\frac {1}{3} A \,b^{2} d^{3} x^{3}\) | \(292\) |
| parallelrisch | \(\frac {1}{9} B \,c^{2} e^{3} x^{9}+\frac {1}{8} x^{8} A \,c^{2} e^{3}+\frac {1}{4} x^{8} B \,e^{3} b c +\frac {3}{8} x^{8} B \,c^{2} d \,e^{2}+\frac {2}{7} x^{7} A b c \,e^{3}+\frac {3}{7} x^{7} A \,c^{2} d \,e^{2}+\frac {1}{7} x^{7} B \,e^{3} b^{2}+\frac {6}{7} x^{7} B b c d \,e^{2}+\frac {3}{7} x^{7} B \,c^{2} d^{2} e +\frac {1}{6} x^{6} A \,b^{2} e^{3}+x^{6} A b c d \,e^{2}+\frac {1}{2} x^{6} A \,c^{2} d^{2} e +\frac {1}{2} x^{6} B \,b^{2} d \,e^{2}+x^{6} B b c \,d^{2} e +\frac {1}{6} x^{6} B \,c^{2} d^{3}+\frac {3}{5} x^{5} A \,b^{2} d \,e^{2}+\frac {6}{5} x^{5} A b c \,d^{2} e +\frac {1}{5} x^{5} A \,d^{3} c^{2}+\frac {3}{5} x^{5} B \,b^{2} d^{2} e +\frac {2}{5} x^{5} B b c \,d^{3}+\frac {3}{4} x^{4} A \,b^{2} d^{2} e +\frac {1}{2} x^{4} A \,d^{3} b c +\frac {1}{4} x^{4} B \,b^{2} d^{3}+\frac {1}{3} A \,b^{2} d^{3} x^{3}\) | \(292\) |
1/9*B*c^2*e^3*x^9+1/8*((A*e^3+3*B*d*e^2)*c^2+2*B*e^3*b*c)*x^8+1/7*((3*A*d* e^2+3*B*d^2*e)*c^2+2*(A*e^3+3*B*d*e^2)*b*c+B*e^3*b^2)*x^7+1/6*((3*A*d^2*e+ B*d^3)*c^2+2*(3*A*d*e^2+3*B*d^2*e)*b*c+(A*e^3+3*B*d*e^2)*b^2)*x^6+1/5*(A*d ^3*c^2+2*(3*A*d^2*e+B*d^3)*b*c+(3*A*d*e^2+3*B*d^2*e)*b^2)*x^5+1/4*(2*A*d^3 *b*c+(3*A*d^2*e+B*d^3)*b^2)*x^4+1/3*A*b^2*d^3*x^3
Time = 0.25 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.05 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{9} \, B c^{2} e^{3} x^{9} + \frac {1}{3} \, A b^{2} d^{3} x^{3} + \frac {1}{8} \, {\left (3 \, B c^{2} d e^{2} + {\left (2 \, B b c + A c^{2}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, B c^{2} d^{2} e + 3 \, {\left (2 \, B b c + A c^{2}\right )} d e^{2} + {\left (B b^{2} + 2 \, A b c\right )} e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{3} + A b^{2} e^{3} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, A b^{2} d e^{2} + {\left (2 \, B b c + A c^{2}\right )} d^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e\right )} x^{5} + \frac {1}{4} \, {\left (3 \, A b^{2} d^{2} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3}\right )} x^{4} \]
1/9*B*c^2*e^3*x^9 + 1/3*A*b^2*d^3*x^3 + 1/8*(3*B*c^2*d*e^2 + (2*B*b*c + A* c^2)*e^3)*x^8 + 1/7*(3*B*c^2*d^2*e + 3*(2*B*b*c + A*c^2)*d*e^2 + (B*b^2 + 2*A*b*c)*e^3)*x^7 + 1/6*(B*c^2*d^3 + A*b^2*e^3 + 3*(2*B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*x^6 + 1/5*(3*A*b^2*d*e^2 + (2*B*b*c + A*c^2) *d^3 + 3*(B*b^2 + 2*A*b*c)*d^2*e)*x^5 + 1/4*(3*A*b^2*d^2*e + (B*b^2 + 2*A* b*c)*d^3)*x^4
Time = 0.03 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.32 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {A b^{2} d^{3} x^{3}}{3} + \frac {B c^{2} e^{3} x^{9}}{9} + x^{8} \left (\frac {A c^{2} e^{3}}{8} + \frac {B b c e^{3}}{4} + \frac {3 B c^{2} d e^{2}}{8}\right ) + x^{7} \cdot \left (\frac {2 A b c e^{3}}{7} + \frac {3 A c^{2} d e^{2}}{7} + \frac {B b^{2} e^{3}}{7} + \frac {6 B b c d e^{2}}{7} + \frac {3 B c^{2} d^{2} e}{7}\right ) + x^{6} \left (\frac {A b^{2} e^{3}}{6} + A b c d e^{2} + \frac {A c^{2} d^{2} e}{2} + \frac {B b^{2} d e^{2}}{2} + B b c d^{2} e + \frac {B c^{2} d^{3}}{6}\right ) + x^{5} \cdot \left (\frac {3 A b^{2} d e^{2}}{5} + \frac {6 A b c d^{2} e}{5} + \frac {A c^{2} d^{3}}{5} + \frac {3 B b^{2} d^{2} e}{5} + \frac {2 B b c d^{3}}{5}\right ) + x^{4} \cdot \left (\frac {3 A b^{2} d^{2} e}{4} + \frac {A b c d^{3}}{2} + \frac {B b^{2} d^{3}}{4}\right ) \]
A*b**2*d**3*x**3/3 + B*c**2*e**3*x**9/9 + x**8*(A*c**2*e**3/8 + B*b*c*e**3 /4 + 3*B*c**2*d*e**2/8) + x**7*(2*A*b*c*e**3/7 + 3*A*c**2*d*e**2/7 + B*b** 2*e**3/7 + 6*B*b*c*d*e**2/7 + 3*B*c**2*d**2*e/7) + x**6*(A*b**2*e**3/6 + A *b*c*d*e**2 + A*c**2*d**2*e/2 + B*b**2*d*e**2/2 + B*b*c*d**2*e + B*c**2*d* *3/6) + x**5*(3*A*b**2*d*e**2/5 + 6*A*b*c*d**2*e/5 + A*c**2*d**3/5 + 3*B*b **2*d**2*e/5 + 2*B*b*c*d**3/5) + x**4*(3*A*b**2*d**2*e/4 + A*b*c*d**3/2 + B*b**2*d**3/4)
Time = 0.19 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.05 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{9} \, B c^{2} e^{3} x^{9} + \frac {1}{3} \, A b^{2} d^{3} x^{3} + \frac {1}{8} \, {\left (3 \, B c^{2} d e^{2} + {\left (2 \, B b c + A c^{2}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, B c^{2} d^{2} e + 3 \, {\left (2 \, B b c + A c^{2}\right )} d e^{2} + {\left (B b^{2} + 2 \, A b c\right )} e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{3} + A b^{2} e^{3} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, A b^{2} d e^{2} + {\left (2 \, B b c + A c^{2}\right )} d^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e\right )} x^{5} + \frac {1}{4} \, {\left (3 \, A b^{2} d^{2} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3}\right )} x^{4} \]
1/9*B*c^2*e^3*x^9 + 1/3*A*b^2*d^3*x^3 + 1/8*(3*B*c^2*d*e^2 + (2*B*b*c + A* c^2)*e^3)*x^8 + 1/7*(3*B*c^2*d^2*e + 3*(2*B*b*c + A*c^2)*d*e^2 + (B*b^2 + 2*A*b*c)*e^3)*x^7 + 1/6*(B*c^2*d^3 + A*b^2*e^3 + 3*(2*B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*x^6 + 1/5*(3*A*b^2*d*e^2 + (2*B*b*c + A*c^2) *d^3 + 3*(B*b^2 + 2*A*b*c)*d^2*e)*x^5 + 1/4*(3*A*b^2*d^2*e + (B*b^2 + 2*A* b*c)*d^3)*x^4
Time = 0.26 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.28 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{9} \, B c^{2} e^{3} x^{9} + \frac {3}{8} \, B c^{2} d e^{2} x^{8} + \frac {1}{4} \, B b c e^{3} x^{8} + \frac {1}{8} \, A c^{2} e^{3} x^{8} + \frac {3}{7} \, B c^{2} d^{2} e x^{7} + \frac {6}{7} \, B b c d e^{2} x^{7} + \frac {3}{7} \, A c^{2} d e^{2} x^{7} + \frac {1}{7} \, B b^{2} e^{3} x^{7} + \frac {2}{7} \, A b c e^{3} x^{7} + \frac {1}{6} \, B c^{2} d^{3} x^{6} + B b c d^{2} e x^{6} + \frac {1}{2} \, A c^{2} d^{2} e x^{6} + \frac {1}{2} \, B b^{2} d e^{2} x^{6} + A b c d e^{2} x^{6} + \frac {1}{6} \, A b^{2} e^{3} x^{6} + \frac {2}{5} \, B b c d^{3} x^{5} + \frac {1}{5} \, A c^{2} d^{3} x^{5} + \frac {3}{5} \, B b^{2} d^{2} e x^{5} + \frac {6}{5} \, A b c d^{2} e x^{5} + \frac {3}{5} \, A b^{2} d e^{2} x^{5} + \frac {1}{4} \, B b^{2} d^{3} x^{4} + \frac {1}{2} \, A b c d^{3} x^{4} + \frac {3}{4} \, A b^{2} d^{2} e x^{4} + \frac {1}{3} \, A b^{2} d^{3} x^{3} \]
1/9*B*c^2*e^3*x^9 + 3/8*B*c^2*d*e^2*x^8 + 1/4*B*b*c*e^3*x^8 + 1/8*A*c^2*e^ 3*x^8 + 3/7*B*c^2*d^2*e*x^7 + 6/7*B*b*c*d*e^2*x^7 + 3/7*A*c^2*d*e^2*x^7 + 1/7*B*b^2*e^3*x^7 + 2/7*A*b*c*e^3*x^7 + 1/6*B*c^2*d^3*x^6 + B*b*c*d^2*e*x^ 6 + 1/2*A*c^2*d^2*e*x^6 + 1/2*B*b^2*d*e^2*x^6 + A*b*c*d*e^2*x^6 + 1/6*A*b^ 2*e^3*x^6 + 2/5*B*b*c*d^3*x^5 + 1/5*A*c^2*d^3*x^5 + 3/5*B*b^2*d^2*e*x^5 + 6/5*A*b*c*d^2*e*x^5 + 3/5*A*b^2*d*e^2*x^5 + 1/4*B*b^2*d^3*x^4 + 1/2*A*b*c* d^3*x^4 + 3/4*A*b^2*d^2*e*x^4 + 1/3*A*b^2*d^3*x^3
Time = 0.11 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.03 \[ \int (A+B x) (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=x^6\,\left (\frac {B\,b^2\,d\,e^2}{2}+\frac {A\,b^2\,e^3}{6}+B\,b\,c\,d^2\,e+A\,b\,c\,d\,e^2+\frac {B\,c^2\,d^3}{6}+\frac {A\,c^2\,d^2\,e}{2}\right )+x^5\,\left (\frac {3\,B\,b^2\,d^2\,e}{5}+\frac {3\,A\,b^2\,d\,e^2}{5}+\frac {2\,B\,b\,c\,d^3}{5}+\frac {6\,A\,b\,c\,d^2\,e}{5}+\frac {A\,c^2\,d^3}{5}\right )+x^7\,\left (\frac {B\,b^2\,e^3}{7}+\frac {6\,B\,b\,c\,d\,e^2}{7}+\frac {2\,A\,b\,c\,e^3}{7}+\frac {3\,B\,c^2\,d^2\,e}{7}+\frac {3\,A\,c^2\,d\,e^2}{7}\right )+\frac {b\,d^2\,x^4\,\left (3\,A\,b\,e+2\,A\,c\,d+B\,b\,d\right )}{4}+\frac {c\,e^2\,x^8\,\left (A\,c\,e+2\,B\,b\,e+3\,B\,c\,d\right )}{8}+\frac {A\,b^2\,d^3\,x^3}{3}+\frac {B\,c^2\,e^3\,x^9}{9} \]
x^6*((A*b^2*e^3)/6 + (B*c^2*d^3)/6 + (A*c^2*d^2*e)/2 + (B*b^2*d*e^2)/2 + A *b*c*d*e^2 + B*b*c*d^2*e) + x^5*((A*c^2*d^3)/5 + (2*B*b*c*d^3)/5 + (3*A*b^ 2*d*e^2)/5 + (3*B*b^2*d^2*e)/5 + (6*A*b*c*d^2*e)/5) + x^7*((B*b^2*e^3)/7 + (2*A*b*c*e^3)/7 + (3*A*c^2*d*e^2)/7 + (3*B*c^2*d^2*e)/7 + (6*B*b*c*d*e^2) /7) + (b*d^2*x^4*(3*A*b*e + 2*A*c*d + B*b*d))/4 + (c*e^2*x^8*(A*c*e + 2*B* b*e + 3*B*c*d))/8 + (A*b^2*d^3*x^3)/3 + (B*c^2*e^3*x^9)/9