Integrand size = 23, antiderivative size = 310 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=a^3 A d x+\frac {1}{2} a^2 (3 A b d+a B d+a A e) x^2+\frac {1}{3} a \left (a B (3 b d+a e)+3 A \left (b^2 d+a c d+a b e\right )\right ) x^3+\frac {1}{4} \left (3 a B \left (b^2 d+a c d+a b e\right )+A \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right )\right ) x^4+\frac {1}{5} \left (b^3 (B d+A e)+6 a b c (B d+A e)+3 b^2 (A c d+a B e)+3 a c (A c d+a B e)\right ) x^5+\frac {1}{6} \left (b^3 B e+3 b^2 c (B d+A e)+3 a c^2 (B d+A e)+3 b c (A c d+2 a B e)\right ) x^6+\frac {1}{7} c \left (3 b^2 B e+3 b c (B d+A e)+c (A c d+3 a B e)\right ) x^7+\frac {1}{8} c^2 (B c d+3 b B e+A c e) x^8+\frac {1}{9} B c^3 e x^9 \]
a^3*A*d*x+1/2*a^2*(A*a*e+3*A*b*d+B*a*d)*x^2+1/3*a*(a*B*(a*e+3*b*d)+3*A*(a* b*e+a*c*d+b^2*d))*x^3+1/4*(3*a*B*(a*b*e+a*c*d+b^2*d)+A*(3*a^2*c*e+3*a*b^2* e+6*a*b*c*d+b^3*d))*x^4+1/5*(b^3*(A*e+B*d)+6*a*b*c*(A*e+B*d)+3*b^2*(A*c*d+ B*a*e)+3*a*c*(A*c*d+B*a*e))*x^5+1/6*(b^3*B*e+3*b^2*c*(A*e+B*d)+3*a*c^2*(A* e+B*d)+3*b*c*(A*c*d+2*B*a*e))*x^6+1/7*c*(3*b^2*B*e+3*b*c*(A*e+B*d)+c*(A*c* d+3*B*a*e))*x^7+1/8*c^2*(A*c*e+3*B*b*e+B*c*d)*x^8+1/9*B*c^3*e*x^9
Time = 0.10 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=a^3 A d x+\frac {1}{2} a^2 (3 A b d+a B d+a A e) x^2+\frac {1}{3} a \left (a B (3 b d+a e)+3 A \left (b^2 d+a c d+a b e\right )\right ) x^3+\frac {1}{4} \left (3 a B \left (b^2 d+a c d+a b e\right )+A \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right )\right ) x^4+\frac {1}{5} \left (b^3 (B d+A e)+6 a b c (B d+A e)+3 b^2 (A c d+a B e)+3 a c (A c d+a B e)\right ) x^5+\frac {1}{6} \left (b^3 B e+3 b^2 c (B d+A e)+3 a c^2 (B d+A e)+3 b c (A c d+2 a B e)\right ) x^6+\frac {1}{7} c \left (3 b^2 B e+3 b c (B d+A e)+c (A c d+3 a B e)\right ) x^7+\frac {1}{8} c^2 (B c d+3 b B e+A c e) x^8+\frac {1}{9} B c^3 e x^9 \]
a^3*A*d*x + (a^2*(3*A*b*d + a*B*d + a*A*e)*x^2)/2 + (a*(a*B*(3*b*d + a*e) + 3*A*(b^2*d + a*c*d + a*b*e))*x^3)/3 + ((3*a*B*(b^2*d + a*c*d + a*b*e) + A*(b^3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c*e))*x^4)/4 + ((b^3*(B*d + A*e) + 6*a*b*c*(B*d + A*e) + 3*b^2*(A*c*d + a*B*e) + 3*a*c*(A*c*d + a*B*e))*x^5 )/5 + ((b^3*B*e + 3*b^2*c*(B*d + A*e) + 3*a*c^2*(B*d + A*e) + 3*b*c*(A*c*d + 2*a*B*e))*x^6)/6 + (c*(3*b^2*B*e + 3*b*c*(B*d + A*e) + c*(A*c*d + 3*a*B *e))*x^7)/7 + (c^2*(B*c*d + 3*b*B*e + A*c*e)*x^8)/8 + (B*c^3*e*x^9)/9
Time = 0.78 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (a^3 A d+x^3 \left (A \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+3 a B \left (a b e+a c d+b^2 d\right )\right )+a^2 x (a A e+a B d+3 A b d)+c x^6 \left (c (3 a B e+A c d)+3 b c (A e+B d)+3 b^2 B e\right )+a x^2 \left (3 A \left (a b e+a c d+b^2 d\right )+a B (a e+3 b d)\right )+x^5 \left (3 b c (2 a B e+A c d)+3 a c^2 (A e+B d)+3 b^2 c (A e+B d)+b^3 B e\right )+x^4 \left (3 b^2 (a B e+A c d)+6 a b c (A e+B d)+3 a c (a B e+A c d)+b^3 (A e+B d)\right )+c^2 x^7 (A c e+3 b B e+B c d)+B c^3 e x^8\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^3 A d x+\frac {1}{4} x^4 \left (A \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+3 a B \left (a b e+a c d+b^2 d\right )\right )+\frac {1}{2} a^2 x^2 (a A e+a B d+3 A b d)+\frac {1}{7} c x^7 \left (c (3 a B e+A c d)+3 b c (A e+B d)+3 b^2 B e\right )+\frac {1}{3} a x^3 \left (3 A \left (a b e+a c d+b^2 d\right )+a B (a e+3 b d)\right )+\frac {1}{6} x^6 \left (3 b c (2 a B e+A c d)+3 a c^2 (A e+B d)+3 b^2 c (A e+B d)+b^3 B e\right )+\frac {1}{5} x^5 \left (3 b^2 (a B e+A c d)+6 a b c (A e+B d)+3 a c (a B e+A c d)+b^3 (A e+B d)\right )+\frac {1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac {1}{9} B c^3 e x^9\) |
a^3*A*d*x + (a^2*(3*A*b*d + a*B*d + a*A*e)*x^2)/2 + (a*(a*B*(3*b*d + a*e) + 3*A*(b^2*d + a*c*d + a*b*e))*x^3)/3 + ((3*a*B*(b^2*d + a*c*d + a*b*e) + A*(b^3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c*e))*x^4)/4 + ((b^3*(B*d + A*e) + 6*a*b*c*(B*d + A*e) + 3*b^2*(A*c*d + a*B*e) + 3*a*c*(A*c*d + a*B*e))*x^5 )/5 + ((b^3*B*e + 3*b^2*c*(B*d + A*e) + 3*a*c^2*(B*d + A*e) + 3*b*c*(A*c*d + 2*a*B*e))*x^6)/6 + (c*(3*b^2*B*e + 3*b*c*(B*d + A*e) + c*(A*c*d + 3*a*B *e))*x^7)/7 + (c^2*(B*c*d + 3*b*B*e + A*c*e)*x^8)/8 + (B*c^3*e*x^9)/9
3.24.37.3.1 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 = 0.34 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.09
| method | result | size |
| norman | \(\frac {B \,c^{3} e \,x^{9}}{9}+\left (\frac {1}{8} A \,c^{3} e +\frac {3}{8} B e b \,c^{2}+\frac {1}{8} B \,c^{3} d \right ) x^{8}+\left (\frac {3}{7} A b \,c^{2} e +\frac {1}{7} A d \,c^{3}+\frac {3}{7} B a \,c^{2} e +\frac {3}{7} B e \,b^{2} c +\frac {3}{7} B b \,c^{2} d \right ) x^{7}+\left (\frac {1}{2} A a \,c^{2} e +\frac {1}{2} A \,b^{2} c e +\frac {1}{2} A d b \,c^{2}+B a b c e +\frac {1}{2} B a \,c^{2} d +\frac {1}{6} b^{3} B e +\frac {1}{2} B \,b^{2} c d \right ) x^{6}+\left (\frac {6}{5} A a b c e +\frac {3}{5} A a \,c^{2} d +\frac {1}{5} A \,b^{3} e +\frac {3}{5} A d \,b^{2} c +\frac {3}{5} B \,a^{2} c e +\frac {3}{5} B e \,b^{2} a +\frac {6}{5} B a b c d +\frac {1}{5} B \,b^{3} d \right ) x^{5}+\left (\frac {3}{4} A \,a^{2} c e +\frac {3}{4} A a \,b^{2} e +\frac {3}{2} A a b c d +\frac {1}{4} A d \,b^{3}+\frac {3}{4} B e b \,a^{2}+\frac {3}{4} B \,a^{2} c d +\frac {3}{4} B a \,b^{2} d \right ) x^{4}+\left (A \,a^{2} b e +A \,a^{2} c d +d A \,b^{2} a +\frac {1}{3} B e \,a^{3}+B \,a^{2} b d \right ) x^{3}+\left (\frac {1}{2} A \,a^{3} e +\frac {3}{2} d A b \,a^{2}+\frac {1}{2} B \,a^{3} d \right ) x^{2}+a^{3} A d x\) | \(339\) |
| default | \(\frac {B \,c^{3} e \,x^{9}}{9}+\frac {\left (\left (A e +B d \right ) c^{3}+3 B e b \,c^{2}\right ) x^{8}}{8}+\frac {\left (A d \,c^{3}+3 \left (A e +B d \right ) b \,c^{2}+B e \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (3 A d b \,c^{2}+\left (A e +B d \right ) \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+B e \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (d A \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+\left (A e +B d \right ) \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+B e \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )\right ) x^{5}}{5}+\frac {\left (d A \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+\left (A e +B d \right ) \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )+3 B e b \,a^{2}\right ) x^{4}}{4}+\frac {\left (d A \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )+3 \left (A e +B d \right ) b \,a^{2}+B e \,a^{3}\right ) x^{3}}{3}+\frac {\left (3 d A b \,a^{2}+\left (A e +B d \right ) a^{3}\right ) x^{2}}{2}+a^{3} A d x\) | \(375\) |
| gosper | \(\frac {3}{7} x^{7} B e a \,c^{2}+\frac {1}{2} x^{6} A a \,c^{2} e +\frac {1}{4} A \,b^{3} d \,x^{4}+\frac {1}{9} B \,c^{3} e \,x^{9}+\frac {3}{8} x^{8} B e b \,c^{2}+\frac {3}{7} x^{7} A b \,c^{2} e +\frac {3}{7} x^{7} B e \,b^{2} c +a^{3} A d x +\frac {1}{5} x^{5} B \,b^{3} d +\frac {1}{5} x^{5} A \,b^{3} e +\frac {1}{7} x^{7} A d \,c^{3}+\frac {1}{6} x^{6} b^{3} B e +\frac {1}{8} x^{8} A \,c^{3} e +\frac {1}{8} x^{8} B \,c^{3} d +\frac {1}{2} x^{2} B \,a^{3} d +\frac {1}{3} x^{3} B e \,a^{3}+\frac {1}{2} x^{2} A \,a^{3} e +\frac {3}{2} x^{4} A a b c d +\frac {6}{5} x^{5} B a b c d +x^{6} B a b c e +\frac {6}{5} x^{5} A a b c e +\frac {3}{5} x^{5} A d \,b^{2} c +\frac {3}{5} x^{5} B e \,b^{2} a +\frac {3}{4} x^{4} A a \,b^{2} e +\frac {3}{4} x^{4} B e b \,a^{2}+\frac {3}{4} x^{4} B a \,b^{2} d +x^{3} A \,a^{2} b e +x^{3} d A \,b^{2} a +x^{3} B \,a^{2} b d +\frac {3}{2} x^{2} d A b \,a^{2}+\frac {1}{2} x^{6} B \,b^{2} c d +\frac {1}{2} x^{6} B a \,c^{2} d +\frac {3}{5} x^{5} d A a \,c^{2}+\frac {3}{5} x^{5} B e c \,a^{2}+\frac {3}{4} x^{4} A \,a^{2} c e +\frac {3}{4} x^{4} B \,a^{2} c d +x^{3} d A c \,a^{2}+\frac {3}{7} x^{7} B b \,c^{2} d +\frac {1}{2} x^{6} A \,b^{2} c e +\frac {1}{2} x^{6} A d b \,c^{2}\) | \(418\) |
| risch | \(\frac {3}{7} x^{7} B e a \,c^{2}+\frac {1}{2} x^{6} A a \,c^{2} e +\frac {1}{4} A \,b^{3} d \,x^{4}+\frac {1}{9} B \,c^{3} e \,x^{9}+\frac {3}{8} x^{8} B e b \,c^{2}+\frac {3}{7} x^{7} A b \,c^{2} e +\frac {3}{7} x^{7} B e \,b^{2} c +a^{3} A d x +\frac {1}{5} x^{5} B \,b^{3} d +\frac {1}{5} x^{5} A \,b^{3} e +\frac {1}{7} x^{7} A d \,c^{3}+\frac {1}{6} x^{6} b^{3} B e +\frac {1}{8} x^{8} A \,c^{3} e +\frac {1}{8} x^{8} B \,c^{3} d +\frac {1}{2} x^{2} B \,a^{3} d +\frac {1}{3} x^{3} B e \,a^{3}+\frac {1}{2} x^{2} A \,a^{3} e +\frac {3}{2} x^{4} A a b c d +\frac {6}{5} x^{5} B a b c d +x^{6} B a b c e +\frac {6}{5} x^{5} A a b c e +\frac {3}{5} x^{5} A d \,b^{2} c +\frac {3}{5} x^{5} B e \,b^{2} a +\frac {3}{4} x^{4} A a \,b^{2} e +\frac {3}{4} x^{4} B e b \,a^{2}+\frac {3}{4} x^{4} B a \,b^{2} d +x^{3} A \,a^{2} b e +x^{3} d A \,b^{2} a +x^{3} B \,a^{2} b d +\frac {3}{2} x^{2} d A b \,a^{2}+\frac {1}{2} x^{6} B \,b^{2} c d +\frac {1}{2} x^{6} B a \,c^{2} d +\frac {3}{5} x^{5} d A a \,c^{2}+\frac {3}{5} x^{5} B e c \,a^{2}+\frac {3}{4} x^{4} A \,a^{2} c e +\frac {3}{4} x^{4} B \,a^{2} c d +x^{3} d A c \,a^{2}+\frac {3}{7} x^{7} B b \,c^{2} d +\frac {1}{2} x^{6} A \,b^{2} c e +\frac {1}{2} x^{6} A d b \,c^{2}\) | \(418\) |
| parallelrisch | \(\frac {3}{7} x^{7} B e a \,c^{2}+\frac {1}{2} x^{6} A a \,c^{2} e +\frac {1}{4} A \,b^{3} d \,x^{4}+\frac {1}{9} B \,c^{3} e \,x^{9}+\frac {3}{8} x^{8} B e b \,c^{2}+\frac {3}{7} x^{7} A b \,c^{2} e +\frac {3}{7} x^{7} B e \,b^{2} c +a^{3} A d x +\frac {1}{5} x^{5} B \,b^{3} d +\frac {1}{5} x^{5} A \,b^{3} e +\frac {1}{7} x^{7} A d \,c^{3}+\frac {1}{6} x^{6} b^{3} B e +\frac {1}{8} x^{8} A \,c^{3} e +\frac {1}{8} x^{8} B \,c^{3} d +\frac {1}{2} x^{2} B \,a^{3} d +\frac {1}{3} x^{3} B e \,a^{3}+\frac {1}{2} x^{2} A \,a^{3} e +\frac {3}{2} x^{4} A a b c d +\frac {6}{5} x^{5} B a b c d +x^{6} B a b c e +\frac {6}{5} x^{5} A a b c e +\frac {3}{5} x^{5} A d \,b^{2} c +\frac {3}{5} x^{5} B e \,b^{2} a +\frac {3}{4} x^{4} A a \,b^{2} e +\frac {3}{4} x^{4} B e b \,a^{2}+\frac {3}{4} x^{4} B a \,b^{2} d +x^{3} A \,a^{2} b e +x^{3} d A \,b^{2} a +x^{3} B \,a^{2} b d +\frac {3}{2} x^{2} d A b \,a^{2}+\frac {1}{2} x^{6} B \,b^{2} c d +\frac {1}{2} x^{6} B a \,c^{2} d +\frac {3}{5} x^{5} d A a \,c^{2}+\frac {3}{5} x^{5} B e c \,a^{2}+\frac {3}{4} x^{4} A \,a^{2} c e +\frac {3}{4} x^{4} B \,a^{2} c d +x^{3} d A c \,a^{2}+\frac {3}{7} x^{7} B b \,c^{2} d +\frac {1}{2} x^{6} A \,b^{2} c e +\frac {1}{2} x^{6} A d b \,c^{2}\) | \(418\) |
1/9*B*c^3*e*x^9+(1/8*A*c^3*e+3/8*B*e*b*c^2+1/8*B*c^3*d)*x^8+(3/7*A*b*c^2*e +1/7*A*d*c^3+3/7*B*a*c^2*e+3/7*B*e*b^2*c+3/7*B*b*c^2*d)*x^7+(1/2*A*a*c^2*e +1/2*A*b^2*c*e+1/2*A*d*b*c^2+B*a*b*c*e+1/2*B*a*c^2*d+1/6*b^3*B*e+1/2*B*b^2 *c*d)*x^6+(6/5*A*a*b*c*e+3/5*A*a*c^2*d+1/5*A*b^3*e+3/5*A*d*b^2*c+3/5*B*a^2 *c*e+3/5*B*e*b^2*a+6/5*B*a*b*c*d+1/5*B*b^3*d)*x^5+(3/4*A*a^2*c*e+3/4*A*a*b ^2*e+3/2*A*a*b*c*d+1/4*A*d*b^3+3/4*B*e*b*a^2+3/4*B*a^2*c*d+3/4*B*a*b^2*d)* x^4+(A*a^2*b*e+A*a^2*c*d+d*A*b^2*a+1/3*B*e*a^3+B*a^2*b*d)*x^3+(1/2*A*a^3*e +3/2*d*A*b*a^2+1/2*B*a^3*d)*x^2+a^3*A*d*x
Time = 0.31 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.08 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{8} \, {\left (B c^{3} d + {\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e\right )} x^{6} + A a^{3} d x + \frac {1}{5} \, {\left ({\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{2} \]
1/9*B*c^3*e*x^9 + 1/8*(B*c^3*d + (3*B*b*c^2 + A*c^3)*e)*x^8 + 1/7*((3*B*b* c^2 + A*c^3)*d + 3*(B*b^2*c + (B*a + A*b)*c^2)*e)*x^7 + 1/6*(3*(B*b^2*c + (B*a + A*b)*c^2)*d + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e)*x^6 + A*a^3*d*x + 1/5*((B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d + (3*B*a*b^ 2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e)*x^5 + 1/4*((3*B*a*b^2 + A*b^3 + 3*(B *a^2 + 2*A*a*b)*c)*d + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*e)*x^4 + 1/3*(3*(B* a^2*b + A*a*b^2 + A*a^2*c)*d + (B*a^3 + 3*A*a^2*b)*e)*x^3 + 1/2*(A*a^3*e + (B*a^3 + 3*A*a^2*b)*d)*x^2
Time = 0.04 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.40 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=A a^{3} d x + \frac {B c^{3} e x^{9}}{9} + x^{8} \left (\frac {A c^{3} e}{8} + \frac {3 B b c^{2} e}{8} + \frac {B c^{3} d}{8}\right ) + x^{7} \cdot \left (\frac {3 A b c^{2} e}{7} + \frac {A c^{3} d}{7} + \frac {3 B a c^{2} e}{7} + \frac {3 B b^{2} c e}{7} + \frac {3 B b c^{2} d}{7}\right ) + x^{6} \left (\frac {A a c^{2} e}{2} + \frac {A b^{2} c e}{2} + \frac {A b c^{2} d}{2} + B a b c e + \frac {B a c^{2} d}{2} + \frac {B b^{3} e}{6} + \frac {B b^{2} c d}{2}\right ) + x^{5} \cdot \left (\frac {6 A a b c e}{5} + \frac {3 A a c^{2} d}{5} + \frac {A b^{3} e}{5} + \frac {3 A b^{2} c d}{5} + \frac {3 B a^{2} c e}{5} + \frac {3 B a b^{2} e}{5} + \frac {6 B a b c d}{5} + \frac {B b^{3} d}{5}\right ) + x^{4} \cdot \left (\frac {3 A a^{2} c e}{4} + \frac {3 A a b^{2} e}{4} + \frac {3 A a b c d}{2} + \frac {A b^{3} d}{4} + \frac {3 B a^{2} b e}{4} + \frac {3 B a^{2} c d}{4} + \frac {3 B a b^{2} d}{4}\right ) + x^{3} \left (A a^{2} b e + A a^{2} c d + A a b^{2} d + \frac {B a^{3} e}{3} + B a^{2} b d\right ) + x^{2} \left (\frac {A a^{3} e}{2} + \frac {3 A a^{2} b d}{2} + \frac {B a^{3} d}{2}\right ) \]
A*a**3*d*x + B*c**3*e*x**9/9 + x**8*(A*c**3*e/8 + 3*B*b*c**2*e/8 + B*c**3* d/8) + x**7*(3*A*b*c**2*e/7 + A*c**3*d/7 + 3*B*a*c**2*e/7 + 3*B*b**2*c*e/7 + 3*B*b*c**2*d/7) + x**6*(A*a*c**2*e/2 + A*b**2*c*e/2 + A*b*c**2*d/2 + B* a*b*c*e + B*a*c**2*d/2 + B*b**3*e/6 + B*b**2*c*d/2) + x**5*(6*A*a*b*c*e/5 + 3*A*a*c**2*d/5 + A*b**3*e/5 + 3*A*b**2*c*d/5 + 3*B*a**2*c*e/5 + 3*B*a*b* *2*e/5 + 6*B*a*b*c*d/5 + B*b**3*d/5) + x**4*(3*A*a**2*c*e/4 + 3*A*a*b**2*e /4 + 3*A*a*b*c*d/2 + A*b**3*d/4 + 3*B*a**2*b*e/4 + 3*B*a**2*c*d/4 + 3*B*a* b**2*d/4) + x**3*(A*a**2*b*e + A*a**2*c*d + A*a*b**2*d + B*a**3*e/3 + B*a* *2*b*d) + x**2*(A*a**3*e/2 + 3*A*a**2*b*d/2 + B*a**3*d/2)
Time = 0.18 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.08 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{8} \, {\left (B c^{3} d + {\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e\right )} x^{6} + A a^{3} d x + \frac {1}{5} \, {\left ({\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{2} \]
1/9*B*c^3*e*x^9 + 1/8*(B*c^3*d + (3*B*b*c^2 + A*c^3)*e)*x^8 + 1/7*((3*B*b* c^2 + A*c^3)*d + 3*(B*b^2*c + (B*a + A*b)*c^2)*e)*x^7 + 1/6*(3*(B*b^2*c + (B*a + A*b)*c^2)*d + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e)*x^6 + A*a^3*d*x + 1/5*((B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d + (3*B*a*b^ 2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e)*x^5 + 1/4*((3*B*a*b^2 + A*b^3 + 3*(B *a^2 + 2*A*a*b)*c)*d + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*e)*x^4 + 1/3*(3*(B* a^2*b + A*a*b^2 + A*a^2*c)*d + (B*a^3 + 3*A*a^2*b)*e)*x^3 + 1/2*(A*a^3*e + (B*a^3 + 3*A*a^2*b)*d)*x^2
Time = 0.27 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.35 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{8} \, B c^{3} d x^{8} + \frac {3}{8} \, B b c^{2} e x^{8} + \frac {1}{8} \, A c^{3} e x^{8} + \frac {3}{7} \, B b c^{2} d x^{7} + \frac {1}{7} \, A c^{3} d x^{7} + \frac {3}{7} \, B b^{2} c e x^{7} + \frac {3}{7} \, B a c^{2} e x^{7} + \frac {3}{7} \, A b c^{2} e x^{7} + \frac {1}{2} \, B b^{2} c d x^{6} + \frac {1}{2} \, B a c^{2} d x^{6} + \frac {1}{2} \, A b c^{2} d x^{6} + \frac {1}{6} \, B b^{3} e x^{6} + B a b c e x^{6} + \frac {1}{2} \, A b^{2} c e x^{6} + \frac {1}{2} \, A a c^{2} e x^{6} + \frac {1}{5} \, B b^{3} d x^{5} + \frac {6}{5} \, B a b c d x^{5} + \frac {3}{5} \, A b^{2} c d x^{5} + \frac {3}{5} \, A a c^{2} d x^{5} + \frac {3}{5} \, B a b^{2} e x^{5} + \frac {1}{5} \, A b^{3} e x^{5} + \frac {3}{5} \, B a^{2} c e x^{5} + \frac {6}{5} \, A a b c e x^{5} + \frac {3}{4} \, B a b^{2} d x^{4} + \frac {1}{4} \, A b^{3} d x^{4} + \frac {3}{4} \, B a^{2} c d x^{4} + \frac {3}{2} \, A a b c d x^{4} + \frac {3}{4} \, B a^{2} b e x^{4} + \frac {3}{4} \, A a b^{2} e x^{4} + \frac {3}{4} \, A a^{2} c e x^{4} + B a^{2} b d x^{3} + A a b^{2} d x^{3} + A a^{2} c d x^{3} + \frac {1}{3} \, B a^{3} e x^{3} + A a^{2} b e x^{3} + \frac {1}{2} \, B a^{3} d x^{2} + \frac {3}{2} \, A a^{2} b d x^{2} + \frac {1}{2} \, A a^{3} e x^{2} + A a^{3} d x \]
1/9*B*c^3*e*x^9 + 1/8*B*c^3*d*x^8 + 3/8*B*b*c^2*e*x^8 + 1/8*A*c^3*e*x^8 + 3/7*B*b*c^2*d*x^7 + 1/7*A*c^3*d*x^7 + 3/7*B*b^2*c*e*x^7 + 3/7*B*a*c^2*e*x^ 7 + 3/7*A*b*c^2*e*x^7 + 1/2*B*b^2*c*d*x^6 + 1/2*B*a*c^2*d*x^6 + 1/2*A*b*c^ 2*d*x^6 + 1/6*B*b^3*e*x^6 + B*a*b*c*e*x^6 + 1/2*A*b^2*c*e*x^6 + 1/2*A*a*c^ 2*e*x^6 + 1/5*B*b^3*d*x^5 + 6/5*B*a*b*c*d*x^5 + 3/5*A*b^2*c*d*x^5 + 3/5*A* a*c^2*d*x^5 + 3/5*B*a*b^2*e*x^5 + 1/5*A*b^3*e*x^5 + 3/5*B*a^2*c*e*x^5 + 6/ 5*A*a*b*c*e*x^5 + 3/4*B*a*b^2*d*x^4 + 1/4*A*b^3*d*x^4 + 3/4*B*a^2*c*d*x^4 + 3/2*A*a*b*c*d*x^4 + 3/4*B*a^2*b*e*x^4 + 3/4*A*a*b^2*e*x^4 + 3/4*A*a^2*c* e*x^4 + B*a^2*b*d*x^3 + A*a*b^2*d*x^3 + A*a^2*c*d*x^3 + 1/3*B*a^3*e*x^3 + A*a^2*b*e*x^3 + 1/2*B*a^3*d*x^2 + 3/2*A*a^2*b*d*x^2 + 1/2*A*a^3*e*x^2 + A* a^3*d*x
Time = 0.13 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.09 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^3 \, dx=x^7\,\left (\frac {A\,c^3\,d}{7}+\frac {3\,A\,b\,c^2\,e}{7}+\frac {3\,B\,a\,c^2\,e}{7}+\frac {3\,B\,b\,c^2\,d}{7}+\frac {3\,B\,b^2\,c\,e}{7}\right )+x^5\,\left (\frac {A\,b^3\,e}{5}+\frac {B\,b^3\,d}{5}+\frac {3\,A\,a\,c^2\,d}{5}+\frac {3\,A\,b^2\,c\,d}{5}+\frac {3\,B\,a\,b^2\,e}{5}+\frac {3\,B\,a^2\,c\,e}{5}+\frac {6\,A\,a\,b\,c\,e}{5}+\frac {6\,B\,a\,b\,c\,d}{5}\right )+x^2\,\left (\frac {A\,a^3\,e}{2}+\frac {B\,a^3\,d}{2}+\frac {3\,A\,a^2\,b\,d}{2}\right )+x^8\,\left (\frac {A\,c^3\,e}{8}+\frac {B\,c^3\,d}{8}+\frac {3\,B\,b\,c^2\,e}{8}\right )+x^4\,\left (\frac {A\,b^3\,d}{4}+\frac {3\,A\,a\,b^2\,e}{4}+\frac {3\,B\,a\,b^2\,d}{4}+\frac {3\,A\,a^2\,c\,e}{4}+\frac {3\,B\,a^2\,b\,e}{4}+\frac {3\,B\,a^2\,c\,d}{4}+\frac {3\,A\,a\,b\,c\,d}{2}\right )+x^6\,\left (\frac {B\,b^3\,e}{6}+\frac {A\,a\,c^2\,e}{2}+\frac {A\,b\,c^2\,d}{2}+\frac {B\,a\,c^2\,d}{2}+\frac {A\,b^2\,c\,e}{2}+\frac {B\,b^2\,c\,d}{2}+B\,a\,b\,c\,e\right )+x^3\,\left (\frac {B\,a^3\,e}{3}+A\,a\,b^2\,d+A\,a^2\,b\,e+A\,a^2\,c\,d+B\,a^2\,b\,d\right )+A\,a^3\,d\,x+\frac {B\,c^3\,e\,x^9}{9} \]
x^7*((A*c^3*d)/7 + (3*A*b*c^2*e)/7 + (3*B*a*c^2*e)/7 + (3*B*b*c^2*d)/7 + ( 3*B*b^2*c*e)/7) + x^5*((A*b^3*e)/5 + (B*b^3*d)/5 + (3*A*a*c^2*d)/5 + (3*A* b^2*c*d)/5 + (3*B*a*b^2*e)/5 + (3*B*a^2*c*e)/5 + (6*A*a*b*c*e)/5 + (6*B*a* b*c*d)/5) + x^2*((A*a^3*e)/2 + (B*a^3*d)/2 + (3*A*a^2*b*d)/2) + x^8*((A*c^ 3*e)/8 + (B*c^3*d)/8 + (3*B*b*c^2*e)/8) + x^4*((A*b^3*d)/4 + (3*A*a*b^2*e) /4 + (3*B*a*b^2*d)/4 + (3*A*a^2*c*e)/4 + (3*B*a^2*b*e)/4 + (3*B*a^2*c*d)/4 + (3*A*a*b*c*d)/2) + x^6*((B*b^3*e)/6 + (A*a*c^2*e)/2 + (A*b*c^2*d)/2 + ( B*a*c^2*d)/2 + (A*b^2*c*e)/2 + (B*b^2*c*d)/2 + B*a*b*c*e) + x^3*((B*a^3*e) /3 + A*a*b^2*d + A*a^2*b*e + A*a^2*c*d + B*a^2*b*d) + A*a^3*d*x + (B*c^3*e *x^9)/9