Integrand size = 24, antiderivative size = 225 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} A b^3 d^2 x^4+\frac {1}{5} b^2 d (b B d+3 A c d+2 A b e) x^5+\frac {1}{6} b \left (3 A c^2 d^2+b^2 e (2 B d+A e)+3 b c d (B d+2 A e)\right ) x^6+\frac {1}{7} \left (A c^3 d^2+b^3 B e^2+3 b^2 c e (2 B d+A e)+3 b c^2 d (B d+2 A e)\right ) x^7+\frac {1}{8} c \left (A c e (2 c d+3 b e)+B \left (c^2 d^2+6 b c d e+3 b^2 e^2\right )\right ) x^8+\frac {1}{9} c^2 e (2 B c d+3 b B e+A c e) x^9+\frac {1}{10} B c^3 e^2 x^{10} \]
1/4*A*b^3*d^2*x^4+1/5*b^2*d*(2*A*b*e+3*A*c*d+B*b*d)*x^5+1/6*b*(3*A*c^2*d^2 +b^2*e*(A*e+2*B*d)+3*b*c*d*(2*A*e+B*d))*x^6+1/7*(A*c^3*d^2+b^3*B*e^2+3*b^2 *c*e*(A*e+2*B*d)+3*b*c^2*d*(2*A*e+B*d))*x^7+1/8*c*(A*c*e*(3*b*e+2*c*d)+B*( 3*b^2*e^2+6*b*c*d*e+c^2*d^2))*x^8+1/9*c^2*e*(A*c*e+3*B*b*e+2*B*c*d)*x^9+1/ 10*B*c^3*e^2*x^10
Time = 0.05 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} A b^3 d^2 x^4+\frac {1}{5} b^2 d (b B d+3 A c d+2 A b e) x^5+\frac {1}{6} b \left (3 A c^2 d^2+b^2 e (2 B d+A e)+3 b c d (B d+2 A e)\right ) x^6+\frac {1}{7} \left (A c^3 d^2+b^3 B e^2+3 b^2 c e (2 B d+A e)+3 b c^2 d (B d+2 A e)\right ) x^7+\frac {1}{8} c \left (A c e (2 c d+3 b e)+B \left (c^2 d^2+6 b c d e+3 b^2 e^2\right )\right ) x^8+\frac {1}{9} c^2 e (2 B c d+3 b B e+A c e) x^9+\frac {1}{10} B c^3 e^2 x^{10} \]
(A*b^3*d^2*x^4)/4 + (b^2*d*(b*B*d + 3*A*c*d + 2*A*b*e)*x^5)/5 + (b*(3*A*c^ 2*d^2 + b^2*e*(2*B*d + A*e) + 3*b*c*d*(B*d + 2*A*e))*x^6)/6 + ((A*c^3*d^2 + b^3*B*e^2 + 3*b^2*c*e*(2*B*d + A*e) + 3*b*c^2*d*(B*d + 2*A*e))*x^7)/7 + (c*(A*c*e*(2*c*d + 3*b*e) + B*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2))*x^8)/8 + (c^2*e*(2*B*c*d + 3*b*B*e + A*c*e)*x^9)/9 + (B*c^3*e^2*x^10)/10
Time = 0.48 (sec) , antiderivative size = 225, 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.
| \(\displaystyle \int (A+B x) \left (b x+c x^2\right )^3 (d+e x)^2 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (A b^3 d^2 x^3+c x^7 \left (A c e (3 b e+2 c d)+B \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+b x^5 \left (b^2 e (A e+2 B d)+3 b c d (2 A e+B d)+3 A c^2 d^2\right )+b^2 d x^4 (2 A b e+3 A c d+b B d)+x^6 \left (3 b^2 c e (A e+2 B d)+3 b c^2 d (2 A e+B d)+A c^3 d^2+b^3 B e^2\right )+c^2 e x^8 (A c e+3 b B e+2 B c d)+B c^3 e^2 x^9\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} A b^3 d^2 x^4+\frac {1}{8} c x^8 \left (A c e (3 b e+2 c d)+B \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac {1}{6} b x^6 \left (b^2 e (A e+2 B d)+3 b c d (2 A e+B d)+3 A c^2 d^2\right )+\frac {1}{5} b^2 d x^5 (2 A b e+3 A c d+b B d)+\frac {1}{7} x^7 \left (3 b^2 c e (A e+2 B d)+3 b c^2 d (2 A e+B d)+A c^3 d^2+b^3 B e^2\right )+\frac {1}{9} c^2 e x^9 (A c e+3 b B e+2 B c d)+\frac {1}{10} B c^3 e^2 x^{10}\) |
(A*b^3*d^2*x^4)/4 + (b^2*d*(b*B*d + 3*A*c*d + 2*A*b*e)*x^5)/5 + (b*(3*A*c^ 2*d^2 + b^2*e*(2*B*d + A*e) + 3*b*c*d*(B*d + 2*A*e))*x^6)/6 + ((A*c^3*d^2 + b^3*B*e^2 + 3*b^2*c*e*(2*B*d + A*e) + 3*b*c^2*d*(B*d + 2*A*e))*x^7)/7 + (c*(A*c*e*(2*c*d + 3*b*e) + B*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2))*x^8)/8 + (c^2*e*(2*B*c*d + 3*b*B*e + A*c*e)*x^9)/9 + (B*c^3*e^2*x^10)/10
3.12.30.3.1 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 = 0.19 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(\frac {B \,c^{3} e^{2} x^{10}}{10}+\frac {\left (\left (A \,e^{2}+2 B d e \right ) c^{3}+3 B \,e^{2} b \,c^{2}\right ) x^{9}}{9}+\frac {\left (\left (2 A d e +B \,d^{2}\right ) c^{3}+3 \left (A \,e^{2}+2 B d e \right ) b \,c^{2}+3 B \,e^{2} b^{2} c \right ) x^{8}}{8}+\frac {\left (A \,c^{3} d^{2}+3 \left (2 A d e +B \,d^{2}\right ) b \,c^{2}+3 \left (A \,e^{2}+2 B d e \right ) b^{2} c +b^{3} B \,e^{2}\right ) x^{7}}{7}+\frac {\left (3 A \,d^{2} b \,c^{2}+3 \left (2 A d e +B \,d^{2}\right ) b^{2} c +\left (A \,e^{2}+2 B d e \right ) b^{3}\right ) x^{6}}{6}+\frac {\left (3 A \,d^{2} b^{2} c +\left (2 A d e +B \,d^{2}\right ) b^{3}\right ) x^{5}}{5}+\frac {A \,b^{3} d^{2} x^{4}}{4}\) | \(240\) |
| norman | \(\frac {B \,c^{3} e^{2} x^{10}}{10}+\left (\frac {1}{9} A \,c^{3} e^{2}+\frac {1}{3} B \,e^{2} b \,c^{2}+\frac {2}{9} B \,c^{3} d e \right ) x^{9}+\left (\frac {3}{8} A b \,c^{2} e^{2}+\frac {1}{4} A \,c^{3} d e +\frac {3}{8} B \,e^{2} b^{2} c +\frac {3}{4} B b \,c^{2} d e +\frac {1}{8} B \,c^{3} d^{2}\right ) x^{8}+\left (\frac {3}{7} A \,b^{2} c \,e^{2}+\frac {6}{7} A b \,c^{2} d e +\frac {1}{7} A \,c^{3} d^{2}+\frac {1}{7} b^{3} B \,e^{2}+\frac {6}{7} B \,b^{2} c d e +\frac {3}{7} B b \,c^{2} d^{2}\right ) x^{7}+\left (\frac {1}{6} A \,b^{3} e^{2}+A \,b^{2} c d e +\frac {1}{2} A \,d^{2} b \,c^{2}+\frac {1}{3} B \,b^{3} d e +\frac {1}{2} B \,b^{2} c \,d^{2}\right ) x^{6}+\left (\frac {2}{5} A \,b^{3} d e +\frac {3}{5} A \,d^{2} b^{2} c +\frac {1}{5} B \,b^{3} d^{2}\right ) x^{5}+\frac {A \,b^{3} d^{2} x^{4}}{4}\) | \(252\) |
| gosper | \(\frac {x^{4} \left (252 B \,c^{3} e^{2} x^{6}+280 x^{5} A \,c^{3} e^{2}+840 x^{5} B \,e^{2} b \,c^{2}+560 x^{5} B \,c^{3} d e +945 x^{4} A b \,c^{2} e^{2}+630 x^{4} A \,c^{3} d e +945 x^{4} B \,e^{2} b^{2} c +1890 x^{4} B b \,c^{2} d e +315 x^{4} B \,c^{3} d^{2}+1080 x^{3} A \,b^{2} c \,e^{2}+2160 x^{3} A b \,c^{2} d e +360 x^{3} A \,c^{3} d^{2}+360 x^{3} b^{3} B \,e^{2}+2160 x^{3} B \,b^{2} c d e +1080 x^{3} B b \,c^{2} d^{2}+420 x^{2} A \,b^{3} e^{2}+2520 x^{2} A \,b^{2} c d e +1260 x^{2} A \,d^{2} b \,c^{2}+840 x^{2} B \,b^{3} d e +1260 x^{2} B \,b^{2} c \,d^{2}+1008 x A \,b^{3} d e +1512 x A \,d^{2} b^{2} c +504 x B \,b^{3} d^{2}+630 A \,d^{2} b^{3}\right )}{2520}\) | \(290\) |
| risch | \(\frac {1}{10} B \,c^{3} e^{2} x^{10}+\frac {1}{9} x^{9} A \,c^{3} e^{2}+\frac {1}{3} x^{9} B \,e^{2} b \,c^{2}+\frac {2}{9} x^{9} B \,c^{3} d e +\frac {3}{8} x^{8} A b \,c^{2} e^{2}+\frac {1}{4} x^{8} A \,c^{3} d e +\frac {3}{8} x^{8} B \,e^{2} b^{2} c +\frac {3}{4} x^{8} B b \,c^{2} d e +\frac {1}{8} x^{8} B \,c^{3} d^{2}+\frac {3}{7} x^{7} A \,b^{2} c \,e^{2}+\frac {6}{7} x^{7} A b \,c^{2} d e +\frac {1}{7} x^{7} A \,c^{3} d^{2}+\frac {1}{7} x^{7} b^{3} B \,e^{2}+\frac {6}{7} x^{7} B \,b^{2} c d e +\frac {3}{7} x^{7} B b \,c^{2} d^{2}+\frac {1}{6} x^{6} A \,b^{3} e^{2}+x^{6} A \,b^{2} c d e +\frac {1}{2} x^{6} A \,d^{2} b \,c^{2}+\frac {1}{3} x^{6} B \,b^{3} d e +\frac {1}{2} x^{6} B \,b^{2} c \,d^{2}+\frac {2}{5} x^{5} A \,b^{3} d e +\frac {3}{5} x^{5} A \,d^{2} b^{2} c +\frac {1}{5} x^{5} B \,b^{3} d^{2}+\frac {1}{4} A \,b^{3} d^{2} x^{4}\) | \(293\) |
| parallelrisch | \(\frac {1}{10} B \,c^{3} e^{2} x^{10}+\frac {1}{9} x^{9} A \,c^{3} e^{2}+\frac {1}{3} x^{9} B \,e^{2} b \,c^{2}+\frac {2}{9} x^{9} B \,c^{3} d e +\frac {3}{8} x^{8} A b \,c^{2} e^{2}+\frac {1}{4} x^{8} A \,c^{3} d e +\frac {3}{8} x^{8} B \,e^{2} b^{2} c +\frac {3}{4} x^{8} B b \,c^{2} d e +\frac {1}{8} x^{8} B \,c^{3} d^{2}+\frac {3}{7} x^{7} A \,b^{2} c \,e^{2}+\frac {6}{7} x^{7} A b \,c^{2} d e +\frac {1}{7} x^{7} A \,c^{3} d^{2}+\frac {1}{7} x^{7} b^{3} B \,e^{2}+\frac {6}{7} x^{7} B \,b^{2} c d e +\frac {3}{7} x^{7} B b \,c^{2} d^{2}+\frac {1}{6} x^{6} A \,b^{3} e^{2}+x^{6} A \,b^{2} c d e +\frac {1}{2} x^{6} A \,d^{2} b \,c^{2}+\frac {1}{3} x^{6} B \,b^{3} d e +\frac {1}{2} x^{6} B \,b^{2} c \,d^{2}+\frac {2}{5} x^{5} A \,b^{3} d e +\frac {3}{5} x^{5} A \,d^{2} b^{2} c +\frac {1}{5} x^{5} B \,b^{3} d^{2}+\frac {1}{4} A \,b^{3} d^{2} x^{4}\) | \(293\) |
1/10*B*c^3*e^2*x^10+1/9*((A*e^2+2*B*d*e)*c^3+3*B*e^2*b*c^2)*x^9+1/8*((2*A* d*e+B*d^2)*c^3+3*(A*e^2+2*B*d*e)*b*c^2+3*B*e^2*b^2*c)*x^8+1/7*(A*c^3*d^2+3 *(2*A*d*e+B*d^2)*b*c^2+3*(A*e^2+2*B*d*e)*b^2*c+b^3*B*e^2)*x^7+1/6*(3*A*d^2 *b*c^2+3*(2*A*d*e+B*d^2)*b^2*c+(A*e^2+2*B*d*e)*b^3)*x^6+1/5*(3*A*d^2*b^2*c +(2*A*d*e+B*d^2)*b^3)*x^5+1/4*A*b^3*d^2*x^4
Time = 0.26 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.08 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=\frac {1}{10} \, B c^{3} e^{2} x^{10} + \frac {1}{4} \, A b^{3} d^{2} x^{4} + \frac {1}{9} \, {\left (2 \, B c^{3} d e + {\left (3 \, B b c^{2} + A c^{3}\right )} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{2} + 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} + 6 \, {\left (B b^{2} c + A b c^{2}\right )} d e + {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (A b^{3} e^{2} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} + 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e\right )} x^{6} + \frac {1}{5} \, {\left (2 \, A b^{3} d e + {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2}\right )} x^{5} \]
1/10*B*c^3*e^2*x^10 + 1/4*A*b^3*d^2*x^4 + 1/9*(2*B*c^3*d*e + (3*B*b*c^2 + A*c^3)*e^2)*x^9 + 1/8*(B*c^3*d^2 + 2*(3*B*b*c^2 + A*c^3)*d*e + 3*(B*b^2*c + A*b*c^2)*e^2)*x^8 + 1/7*((3*B*b*c^2 + A*c^3)*d^2 + 6*(B*b^2*c + A*b*c^2) *d*e + (B*b^3 + 3*A*b^2*c)*e^2)*x^7 + 1/6*(A*b^3*e^2 + 3*(B*b^2*c + A*b*c^ 2)*d^2 + 2*(B*b^3 + 3*A*b^2*c)*d*e)*x^6 + 1/5*(2*A*b^3*d*e + (B*b^3 + 3*A* b^2*c)*d^2)*x^5
Time = 0.04 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.35 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=\frac {A b^{3} d^{2} x^{4}}{4} + \frac {B c^{3} e^{2} x^{10}}{10} + x^{9} \left (\frac {A c^{3} e^{2}}{9} + \frac {B b c^{2} e^{2}}{3} + \frac {2 B c^{3} d e}{9}\right ) + x^{8} \cdot \left (\frac {3 A b c^{2} e^{2}}{8} + \frac {A c^{3} d e}{4} + \frac {3 B b^{2} c e^{2}}{8} + \frac {3 B b c^{2} d e}{4} + \frac {B c^{3} d^{2}}{8}\right ) + x^{7} \cdot \left (\frac {3 A b^{2} c e^{2}}{7} + \frac {6 A b c^{2} d e}{7} + \frac {A c^{3} d^{2}}{7} + \frac {B b^{3} e^{2}}{7} + \frac {6 B b^{2} c d e}{7} + \frac {3 B b c^{2} d^{2}}{7}\right ) + x^{6} \left (\frac {A b^{3} e^{2}}{6} + A b^{2} c d e + \frac {A b c^{2} d^{2}}{2} + \frac {B b^{3} d e}{3} + \frac {B b^{2} c d^{2}}{2}\right ) + x^{5} \cdot \left (\frac {2 A b^{3} d e}{5} + \frac {3 A b^{2} c d^{2}}{5} + \frac {B b^{3} d^{2}}{5}\right ) \]
A*b**3*d**2*x**4/4 + B*c**3*e**2*x**10/10 + x**9*(A*c**3*e**2/9 + B*b*c**2 *e**2/3 + 2*B*c**3*d*e/9) + x**8*(3*A*b*c**2*e**2/8 + A*c**3*d*e/4 + 3*B*b **2*c*e**2/8 + 3*B*b*c**2*d*e/4 + B*c**3*d**2/8) + x**7*(3*A*b**2*c*e**2/7 + 6*A*b*c**2*d*e/7 + A*c**3*d**2/7 + B*b**3*e**2/7 + 6*B*b**2*c*d*e/7 + 3 *B*b*c**2*d**2/7) + x**6*(A*b**3*e**2/6 + A*b**2*c*d*e + A*b*c**2*d**2/2 + B*b**3*d*e/3 + B*b**2*c*d**2/2) + x**5*(2*A*b**3*d*e/5 + 3*A*b**2*c*d**2/ 5 + B*b**3*d**2/5)
Time = 0.21 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.08 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=\frac {1}{10} \, B c^{3} e^{2} x^{10} + \frac {1}{4} \, A b^{3} d^{2} x^{4} + \frac {1}{9} \, {\left (2 \, B c^{3} d e + {\left (3 \, B b c^{2} + A c^{3}\right )} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{2} + 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} + 6 \, {\left (B b^{2} c + A b c^{2}\right )} d e + {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (A b^{3} e^{2} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} + 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e\right )} x^{6} + \frac {1}{5} \, {\left (2 \, A b^{3} d e + {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2}\right )} x^{5} \]
1/10*B*c^3*e^2*x^10 + 1/4*A*b^3*d^2*x^4 + 1/9*(2*B*c^3*d*e + (3*B*b*c^2 + A*c^3)*e^2)*x^9 + 1/8*(B*c^3*d^2 + 2*(3*B*b*c^2 + A*c^3)*d*e + 3*(B*b^2*c + A*b*c^2)*e^2)*x^8 + 1/7*((3*B*b*c^2 + A*c^3)*d^2 + 6*(B*b^2*c + A*b*c^2) *d*e + (B*b^3 + 3*A*b^2*c)*e^2)*x^7 + 1/6*(A*b^3*e^2 + 3*(B*b^2*c + A*b*c^ 2)*d^2 + 2*(B*b^3 + 3*A*b^2*c)*d*e)*x^6 + 1/5*(2*A*b^3*d*e + (B*b^3 + 3*A* b^2*c)*d^2)*x^5
Time = 0.26 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.30 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=\frac {1}{10} \, B c^{3} e^{2} x^{10} + \frac {2}{9} \, B c^{3} d e x^{9} + \frac {1}{3} \, B b c^{2} e^{2} x^{9} + \frac {1}{9} \, A c^{3} e^{2} x^{9} + \frac {1}{8} \, B c^{3} d^{2} x^{8} + \frac {3}{4} \, B b c^{2} d e x^{8} + \frac {1}{4} \, A c^{3} d e x^{8} + \frac {3}{8} \, B b^{2} c e^{2} x^{8} + \frac {3}{8} \, A b c^{2} e^{2} x^{8} + \frac {3}{7} \, B b c^{2} d^{2} x^{7} + \frac {1}{7} \, A c^{3} d^{2} x^{7} + \frac {6}{7} \, B b^{2} c d e x^{7} + \frac {6}{7} \, A b c^{2} d e x^{7} + \frac {1}{7} \, B b^{3} e^{2} x^{7} + \frac {3}{7} \, A b^{2} c e^{2} x^{7} + \frac {1}{2} \, B b^{2} c d^{2} x^{6} + \frac {1}{2} \, A b c^{2} d^{2} x^{6} + \frac {1}{3} \, B b^{3} d e x^{6} + A b^{2} c d e x^{6} + \frac {1}{6} \, A b^{3} e^{2} x^{6} + \frac {1}{5} \, B b^{3} d^{2} x^{5} + \frac {3}{5} \, A b^{2} c d^{2} x^{5} + \frac {2}{5} \, A b^{3} d e x^{5} + \frac {1}{4} \, A b^{3} d^{2} x^{4} \]
1/10*B*c^3*e^2*x^10 + 2/9*B*c^3*d*e*x^9 + 1/3*B*b*c^2*e^2*x^9 + 1/9*A*c^3* e^2*x^9 + 1/8*B*c^3*d^2*x^8 + 3/4*B*b*c^2*d*e*x^8 + 1/4*A*c^3*d*e*x^8 + 3/ 8*B*b^2*c*e^2*x^8 + 3/8*A*b*c^2*e^2*x^8 + 3/7*B*b*c^2*d^2*x^7 + 1/7*A*c^3* d^2*x^7 + 6/7*B*b^2*c*d*e*x^7 + 6/7*A*b*c^2*d*e*x^7 + 1/7*B*b^3*e^2*x^7 + 3/7*A*b^2*c*e^2*x^7 + 1/2*B*b^2*c*d^2*x^6 + 1/2*A*b*c^2*d^2*x^6 + 1/3*B*b^ 3*d*e*x^6 + A*b^2*c*d*e*x^6 + 1/6*A*b^3*e^2*x^6 + 1/5*B*b^3*d^2*x^5 + 3/5* A*b^2*c*d^2*x^5 + 2/5*A*b^3*d*e*x^5 + 1/4*A*b^3*d^2*x^4
Time = 0.10 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.04 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=x^7\,\left (\frac {B\,b^3\,e^2}{7}+\frac {6\,B\,b^2\,c\,d\,e}{7}+\frac {3\,A\,b^2\,c\,e^2}{7}+\frac {3\,B\,b\,c^2\,d^2}{7}+\frac {6\,A\,b\,c^2\,d\,e}{7}+\frac {A\,c^3\,d^2}{7}\right )+x^6\,\left (\frac {B\,b^3\,d\,e}{3}+\frac {A\,b^3\,e^2}{6}+\frac {B\,b^2\,c\,d^2}{2}+A\,b^2\,c\,d\,e+\frac {A\,b\,c^2\,d^2}{2}\right )+x^8\,\left (\frac {3\,B\,b^2\,c\,e^2}{8}+\frac {3\,B\,b\,c^2\,d\,e}{4}+\frac {3\,A\,b\,c^2\,e^2}{8}+\frac {B\,c^3\,d^2}{8}+\frac {A\,c^3\,d\,e}{4}\right )+\frac {b^2\,d\,x^5\,\left (2\,A\,b\,e+3\,A\,c\,d+B\,b\,d\right )}{5}+\frac {c^2\,e\,x^9\,\left (A\,c\,e+3\,B\,b\,e+2\,B\,c\,d\right )}{9}+\frac {A\,b^3\,d^2\,x^4}{4}+\frac {B\,c^3\,e^2\,x^{10}}{10} \]
x^7*((A*c^3*d^2)/7 + (B*b^3*e^2)/7 + (3*A*b^2*c*e^2)/7 + (3*B*b*c^2*d^2)/7 + (6*A*b*c^2*d*e)/7 + (6*B*b^2*c*d*e)/7) + x^6*((A*b^3*e^2)/6 + (B*b^3*d* e)/3 + (A*b*c^2*d^2)/2 + (B*b^2*c*d^2)/2 + A*b^2*c*d*e) + x^8*((B*c^3*d^2) /8 + (A*c^3*d*e)/4 + (3*A*b*c^2*e^2)/8 + (3*B*b^2*c*e^2)/8 + (3*B*b*c^2*d* e)/4) + (b^2*d*x^5*(2*A*b*e + 3*A*c*d + B*b*d))/5 + (c^2*e*x^9*(A*c*e + 3* B*b*e + 2*B*c*d))/9 + (A*b^3*d^2*x^4)/4 + (B*c^3*e^2*x^10)/10