Integrand size = 26, antiderivative size = 240 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}{4 e^6}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^5}{5 e^6}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^6}{6 e^6}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^7}{7 e^6}-\frac {5 c^2 (2 c d-b e) (d+e x)^8}{8 e^6}+\frac {2 c^3 (d+e x)^9}{9 e^6} \]
-1/4*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^4/e^6+2/5*(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)^5/e^6-1/6*(-b*e+2*c*d)*(1 0*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))*(e*x+d)^6/e^6+4/7*c*(5*c^2*d^2+b^2 *e^2-c*e*(-a*e+5*b*d))*(e*x+d)^7/e^6-5/8*c^2*(-b*e+2*c*d)*(e*x+d)^8/e^6+2/ 9*c^3*(e*x+d)^9/e^6
Time = 0.11 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.46 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=a^2 b d^3 x+\frac {1}{2} a d^2 \left (2 b^2 d+2 a c d+3 a b e\right ) x^2+\frac {1}{3} d \left (b^3 d^2+6 a b^2 d e+6 a^2 c d e+3 a b \left (2 c d^2+a e^2\right )\right ) x^3+\frac {1}{4} \left (3 b^3 d^2 e+a b e \left (18 c d^2+a e^2\right )+2 a c d \left (2 c d^2+3 a e^2\right )+b^2 \left (4 c d^3+6 a d e^2\right )\right ) x^4+\frac {1}{5} \left (3 b^3 d e^2+2 a c e \left (6 c d^2+a e^2\right )+b c d \left (5 c d^2+18 a e^2\right )+2 b^2 \left (6 c d^2 e+a e^3\right )\right ) x^5+\frac {1}{6} \left (2 c^3 d^3+b^3 e^3+6 b c e^2 (2 b d+a e)+3 c^2 d e (5 b d+4 a e)\right ) x^6+\frac {1}{7} c e \left (6 c^2 d^2+4 b^2 e^2+c e (15 b d+4 a e)\right ) x^7+\frac {1}{8} c^2 e^2 (6 c d+5 b e) x^8+\frac {2}{9} c^3 e^3 x^9 \]
a^2*b*d^3*x + (a*d^2*(2*b^2*d + 2*a*c*d + 3*a*b*e)*x^2)/2 + (d*(b^3*d^2 + 6*a*b^2*d*e + 6*a^2*c*d*e + 3*a*b*(2*c*d^2 + a*e^2))*x^3)/3 + ((3*b^3*d^2* e + a*b*e*(18*c*d^2 + a*e^2) + 2*a*c*d*(2*c*d^2 + 3*a*e^2) + b^2*(4*c*d^3 + 6*a*d*e^2))*x^4)/4 + ((3*b^3*d*e^2 + 2*a*c*e*(6*c*d^2 + a*e^2) + b*c*d*( 5*c*d^2 + 18*a*e^2) + 2*b^2*(6*c*d^2*e + a*e^3))*x^5)/5 + ((2*c^3*d^3 + b^ 3*e^3 + 6*b*c*e^2*(2*b*d + a*e) + 3*c^2*d*e*(5*b*d + 4*a*e))*x^6)/6 + (c*e *(6*c^2*d^2 + 4*b^2*e^2 + c*e*(15*b*d + 4*a*e))*x^7)/7 + (c^2*e^2*(6*c*d + 5*b*e)*x^8)/8 + (2*c^3*e^3*x^9)/9
Time = 0.58 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
| \(\displaystyle \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {4 c (d+e x)^6 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^5}+\frac {(d+e x)^5 (2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^5}+\frac {2 (d+e x)^4 \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^5}+\frac {(d+e x)^3 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^5}-\frac {5 c^2 (d+e x)^7 (2 c d-b e)}{e^5}+\frac {2 c^3 (d+e x)^8}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 c (d+e x)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac {(d+e x)^6 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^6}+\frac {2 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6}-\frac {5 c^2 (d+e x)^8 (2 c d-b e)}{8 e^6}+\frac {2 c^3 (d+e x)^9}{9 e^6}\) |
-1/4*((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^4)/e^6 + (2*(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)^5)/( 5*e^6) - ((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)^6)/(6*e^6) + (4*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e* x)^7)/(7*e^6) - (5*c^2*(2*c*d - b*e)*(d + e*x)^8)/(8*e^6) + (2*c^3*(d + e* x)^9)/(9*e^6)
3.16.5.3.1 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 = 0.39 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.51
| method | result | size |
| norman | \(\frac {2 c^{3} e^{3} x^{9}}{9}+\left (\frac {5}{8} c^{2} e^{3} b +\frac {3}{4} c^{3} d \,e^{2}\right ) x^{8}+\left (\frac {4}{7} a \,c^{2} e^{3}+\frac {4}{7} b^{2} c \,e^{3}+\frac {15}{7} b \,c^{2} d \,e^{2}+\frac {6}{7} c^{3} d^{2} e \right ) x^{7}+\left (c \,e^{3} b a +2 a \,c^{2} d \,e^{2}+\frac {1}{6} b^{3} e^{3}+2 b^{2} c d \,e^{2}+\frac {5}{2} b \,c^{2} d^{2} e +\frac {1}{3} c^{3} d^{3}\right ) x^{6}+\left (\frac {2}{5} c \,e^{3} a^{2}+\frac {2}{5} a \,b^{2} e^{3}+\frac {18}{5} a b c d \,e^{2}+\frac {12}{5} a \,c^{2} d^{2} e +\frac {3}{5} b^{3} d \,e^{2}+\frac {12}{5} b^{2} c \,d^{2} e +b \,d^{3} c^{2}\right ) x^{5}+\left (\frac {1}{4} a^{2} b \,e^{3}+\frac {3}{2} a^{2} c d \,e^{2}+\frac {3}{2} a \,b^{2} d \,e^{2}+\frac {9}{2} a b c \,d^{2} e +a \,c^{2} d^{3}+\frac {3}{4} b^{3} d^{2} e +b^{2} d^{3} c \right ) x^{4}+\left (a^{2} b d \,e^{2}+2 a^{2} c \,d^{2} e +2 a \,b^{2} d^{2} e +2 a b c \,d^{3}+\frac {1}{3} b^{3} d^{3}\right ) x^{3}+\left (\frac {3}{2} a^{2} b \,d^{2} e +a^{2} c \,d^{3}+b^{2} d^{3} a \right ) x^{2}+b \,d^{3} a^{2} x\) | \(363\) |
| default | \(\frac {2 c^{3} e^{3} x^{9}}{9}+\frac {\left (\left (b \,e^{3}+6 c d \,e^{2}\right ) c^{2}+4 c^{2} e^{3} b \right ) x^{8}}{8}+\frac {\left (\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) c^{2}+2 \left (b \,e^{3}+6 c d \,e^{2}\right ) b c +2 c \,e^{3} \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (\left (3 b \,d^{2} e +2 c \,d^{3}\right ) c^{2}+2 \left (3 b d \,e^{2}+6 c \,d^{2} e \right ) b c +\left (b \,e^{3}+6 c d \,e^{2}\right ) \left (2 a c +b^{2}\right )+4 c \,e^{3} b a \right ) x^{6}}{6}+\frac {\left (b \,d^{3} c^{2}+2 \left (3 b \,d^{2} e +2 c \,d^{3}\right ) b c +\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) \left (2 a c +b^{2}\right )+2 \left (b \,e^{3}+6 c d \,e^{2}\right ) b a +2 c \,e^{3} a^{2}\right ) x^{5}}{5}+\frac {\left (2 b^{2} d^{3} c +\left (3 b \,d^{2} e +2 c \,d^{3}\right ) \left (2 a c +b^{2}\right )+2 \left (3 b d \,e^{2}+6 c \,d^{2} e \right ) b a +\left (b \,e^{3}+6 c d \,e^{2}\right ) a^{2}\right ) x^{4}}{4}+\frac {\left (b \,d^{3} \left (2 a c +b^{2}\right )+2 \left (3 b \,d^{2} e +2 c \,d^{3}\right ) b a +\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) a^{2}\right ) x^{3}}{3}+\frac {\left (2 b^{2} d^{3} a +\left (3 b \,d^{2} e +2 c \,d^{3}\right ) a^{2}\right ) x^{2}}{2}+b \,d^{3} a^{2} x\) | \(428\) |
| gosper | \(\frac {2}{9} c^{3} e^{3} x^{9}+\frac {1}{6} x^{6} b^{3} e^{3}+\frac {1}{3} x^{6} c^{3} d^{3}+\frac {1}{3} x^{3} b^{3} d^{3}+\frac {3}{4} x^{4} b^{3} d^{2} e +x^{4} b^{2} d^{3} c +x^{2} a^{2} c \,d^{3}+x^{2} b^{2} d^{3} a +x^{5} b \,d^{3} c^{2}+\frac {1}{4} x^{4} a^{2} b \,e^{3}+x^{4} a \,c^{2} d^{3}+\frac {2}{5} x^{5} a \,b^{2} e^{3}+\frac {3}{5} x^{5} b^{3} d \,e^{2}+\frac {6}{7} x^{7} c^{3} d^{2} e +\frac {2}{5} x^{5} c \,e^{3} a^{2}+\frac {4}{7} x^{7} a \,c^{2} e^{3}+\frac {4}{7} x^{7} b^{2} c \,e^{3}+\frac {5}{8} x^{8} c^{2} e^{3} b +\frac {3}{4} x^{8} c^{3} d \,e^{2}+b \,d^{3} a^{2} x +\frac {9}{2} x^{4} a b c \,d^{2} e +\frac {18}{5} x^{5} a b c d \,e^{2}+\frac {15}{7} x^{7} b \,c^{2} d \,e^{2}+x^{6} c \,e^{3} b a +2 x^{6} a \,c^{2} d \,e^{2}+2 x^{6} b^{2} c d \,e^{2}+\frac {5}{2} x^{6} b \,c^{2} d^{2} e +\frac {12}{5} x^{5} a \,c^{2} d^{2} e +\frac {12}{5} x^{5} b^{2} c \,d^{2} e +\frac {3}{2} x^{4} a^{2} c d \,e^{2}+\frac {3}{2} x^{4} a \,b^{2} d \,e^{2}+x^{3} a^{2} b d \,e^{2}+2 x^{3} a^{2} c \,d^{2} e +2 x^{3} a \,b^{2} d^{2} e +2 x^{3} a b c \,d^{3}+\frac {3}{2} x^{2} a^{2} b \,d^{2} e\) | \(430\) |
| risch | \(\frac {2}{9} c^{3} e^{3} x^{9}+\frac {1}{6} x^{6} b^{3} e^{3}+\frac {1}{3} x^{6} c^{3} d^{3}+\frac {1}{3} x^{3} b^{3} d^{3}+\frac {3}{4} x^{4} b^{3} d^{2} e +x^{4} b^{2} d^{3} c +x^{2} a^{2} c \,d^{3}+x^{2} b^{2} d^{3} a +x^{5} b \,d^{3} c^{2}+\frac {1}{4} x^{4} a^{2} b \,e^{3}+x^{4} a \,c^{2} d^{3}+\frac {2}{5} x^{5} a \,b^{2} e^{3}+\frac {3}{5} x^{5} b^{3} d \,e^{2}+\frac {6}{7} x^{7} c^{3} d^{2} e +\frac {2}{5} x^{5} c \,e^{3} a^{2}+\frac {4}{7} x^{7} a \,c^{2} e^{3}+\frac {4}{7} x^{7} b^{2} c \,e^{3}+\frac {5}{8} x^{8} c^{2} e^{3} b +\frac {3}{4} x^{8} c^{3} d \,e^{2}+b \,d^{3} a^{2} x +\frac {9}{2} x^{4} a b c \,d^{2} e +\frac {18}{5} x^{5} a b c d \,e^{2}+\frac {15}{7} x^{7} b \,c^{2} d \,e^{2}+x^{6} c \,e^{3} b a +2 x^{6} a \,c^{2} d \,e^{2}+2 x^{6} b^{2} c d \,e^{2}+\frac {5}{2} x^{6} b \,c^{2} d^{2} e +\frac {12}{5} x^{5} a \,c^{2} d^{2} e +\frac {12}{5} x^{5} b^{2} c \,d^{2} e +\frac {3}{2} x^{4} a^{2} c d \,e^{2}+\frac {3}{2} x^{4} a \,b^{2} d \,e^{2}+x^{3} a^{2} b d \,e^{2}+2 x^{3} a^{2} c \,d^{2} e +2 x^{3} a \,b^{2} d^{2} e +2 x^{3} a b c \,d^{3}+\frac {3}{2} x^{2} a^{2} b \,d^{2} e\) | \(430\) |
| parallelrisch | \(\frac {2}{9} c^{3} e^{3} x^{9}+\frac {1}{6} x^{6} b^{3} e^{3}+\frac {1}{3} x^{6} c^{3} d^{3}+\frac {1}{3} x^{3} b^{3} d^{3}+\frac {3}{4} x^{4} b^{3} d^{2} e +x^{4} b^{2} d^{3} c +x^{2} a^{2} c \,d^{3}+x^{2} b^{2} d^{3} a +x^{5} b \,d^{3} c^{2}+\frac {1}{4} x^{4} a^{2} b \,e^{3}+x^{4} a \,c^{2} d^{3}+\frac {2}{5} x^{5} a \,b^{2} e^{3}+\frac {3}{5} x^{5} b^{3} d \,e^{2}+\frac {6}{7} x^{7} c^{3} d^{2} e +\frac {2}{5} x^{5} c \,e^{3} a^{2}+\frac {4}{7} x^{7} a \,c^{2} e^{3}+\frac {4}{7} x^{7} b^{2} c \,e^{3}+\frac {5}{8} x^{8} c^{2} e^{3} b +\frac {3}{4} x^{8} c^{3} d \,e^{2}+b \,d^{3} a^{2} x +\frac {9}{2} x^{4} a b c \,d^{2} e +\frac {18}{5} x^{5} a b c d \,e^{2}+\frac {15}{7} x^{7} b \,c^{2} d \,e^{2}+x^{6} c \,e^{3} b a +2 x^{6} a \,c^{2} d \,e^{2}+2 x^{6} b^{2} c d \,e^{2}+\frac {5}{2} x^{6} b \,c^{2} d^{2} e +\frac {12}{5} x^{5} a \,c^{2} d^{2} e +\frac {12}{5} x^{5} b^{2} c \,d^{2} e +\frac {3}{2} x^{4} a^{2} c d \,e^{2}+\frac {3}{2} x^{4} a \,b^{2} d \,e^{2}+x^{3} a^{2} b d \,e^{2}+2 x^{3} a^{2} c \,d^{2} e +2 x^{3} a \,b^{2} d^{2} e +2 x^{3} a b c \,d^{3}+\frac {3}{2} x^{2} a^{2} b \,d^{2} e\) | \(430\) |
2/9*c^3*e^3*x^9+(5/8*c^2*e^3*b+3/4*c^3*d*e^2)*x^8+(4/7*a*c^2*e^3+4/7*b^2*c *e^3+15/7*b*c^2*d*e^2+6/7*c^3*d^2*e)*x^7+(c*e^3*b*a+2*a*c^2*d*e^2+1/6*b^3* e^3+2*b^2*c*d*e^2+5/2*b*c^2*d^2*e+1/3*c^3*d^3)*x^6+(2/5*c*e^3*a^2+2/5*a*b^ 2*e^3+18/5*a*b*c*d*e^2+12/5*a*c^2*d^2*e+3/5*b^3*d*e^2+12/5*b^2*c*d^2*e+b*d ^3*c^2)*x^5+(1/4*a^2*b*e^3+3/2*a^2*c*d*e^2+3/2*a*b^2*d*e^2+9/2*a*b*c*d^2*e +a*c^2*d^3+3/4*b^3*d^2*e+b^2*d^3*c)*x^4+(a^2*b*d*e^2+2*a^2*c*d^2*e+2*a*b^2 *d^2*e+2*a*b*c*d^3+1/3*b^3*d^3)*x^3+(3/2*a^2*b*d^2*e+a^2*c*d^3+b^2*d^3*a)* x^2+b*d^3*a^2*x
Time = 0.38 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.43 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{9} \, c^{3} e^{3} x^{9} + \frac {1}{8} \, {\left (6 \, c^{3} d e^{2} + 5 \, b c^{2} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, c^{3} d^{2} e + 15 \, b c^{2} d e^{2} + 4 \, {\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{7} + a^{2} b d^{3} x + \frac {1}{6} \, {\left (2 \, c^{3} d^{3} + 15 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} + {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, b c^{2} d^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e + 3 \, {\left (b^{3} + 6 \, a b c\right )} d e^{2} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (a^{2} b e^{3} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} + 3 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e + 6 \, {\left (a b^{2} + a^{2} c\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a^{2} b d e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{3} + 6 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{2} e + 2 \, {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} x^{2} \]
2/9*c^3*e^3*x^9 + 1/8*(6*c^3*d*e^2 + 5*b*c^2*e^3)*x^8 + 1/7*(6*c^3*d^2*e + 15*b*c^2*d*e^2 + 4*(b^2*c + a*c^2)*e^3)*x^7 + a^2*b*d^3*x + 1/6*(2*c^3*d^ 3 + 15*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 + (b^3 + 6*a*b*c)*e^3)*x^6 + 1/5*(5*b*c^2*d^3 + 12*(b^2*c + a*c^2)*d^2*e + 3*(b^3 + 6*a*b*c)*d*e^2 + 2 *(a*b^2 + a^2*c)*e^3)*x^5 + 1/4*(a^2*b*e^3 + 4*(b^2*c + a*c^2)*d^3 + 3*(b^ 3 + 6*a*b*c)*d^2*e + 6*(a*b^2 + a^2*c)*d*e^2)*x^4 + 1/3*(3*a^2*b*d*e^2 + ( b^3 + 6*a*b*c)*d^3 + 6*(a*b^2 + a^2*c)*d^2*e)*x^3 + 1/2*(3*a^2*b*d^2*e + 2 *(a*b^2 + a^2*c)*d^3)*x^2
Time = 0.04 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.79 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=a^{2} b d^{3} x + \frac {2 c^{3} e^{3} x^{9}}{9} + x^{8} \cdot \left (\frac {5 b c^{2} e^{3}}{8} + \frac {3 c^{3} d e^{2}}{4}\right ) + x^{7} \cdot \left (\frac {4 a c^{2} e^{3}}{7} + \frac {4 b^{2} c e^{3}}{7} + \frac {15 b c^{2} d e^{2}}{7} + \frac {6 c^{3} d^{2} e}{7}\right ) + x^{6} \left (a b c e^{3} + 2 a c^{2} d e^{2} + \frac {b^{3} e^{3}}{6} + 2 b^{2} c d e^{2} + \frac {5 b c^{2} d^{2} e}{2} + \frac {c^{3} d^{3}}{3}\right ) + x^{5} \cdot \left (\frac {2 a^{2} c e^{3}}{5} + \frac {2 a b^{2} e^{3}}{5} + \frac {18 a b c d e^{2}}{5} + \frac {12 a c^{2} d^{2} e}{5} + \frac {3 b^{3} d e^{2}}{5} + \frac {12 b^{2} c d^{2} e}{5} + b c^{2} d^{3}\right ) + x^{4} \left (\frac {a^{2} b e^{3}}{4} + \frac {3 a^{2} c d e^{2}}{2} + \frac {3 a b^{2} d e^{2}}{2} + \frac {9 a b c d^{2} e}{2} + a c^{2} d^{3} + \frac {3 b^{3} d^{2} e}{4} + b^{2} c d^{3}\right ) + x^{3} \left (a^{2} b d e^{2} + 2 a^{2} c d^{2} e + 2 a b^{2} d^{2} e + 2 a b c d^{3} + \frac {b^{3} d^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 a^{2} b d^{2} e}{2} + a^{2} c d^{3} + a b^{2} d^{3}\right ) \]
a**2*b*d**3*x + 2*c**3*e**3*x**9/9 + x**8*(5*b*c**2*e**3/8 + 3*c**3*d*e**2 /4) + x**7*(4*a*c**2*e**3/7 + 4*b**2*c*e**3/7 + 15*b*c**2*d*e**2/7 + 6*c** 3*d**2*e/7) + x**6*(a*b*c*e**3 + 2*a*c**2*d*e**2 + b**3*e**3/6 + 2*b**2*c* d*e**2 + 5*b*c**2*d**2*e/2 + c**3*d**3/3) + x**5*(2*a**2*c*e**3/5 + 2*a*b* *2*e**3/5 + 18*a*b*c*d*e**2/5 + 12*a*c**2*d**2*e/5 + 3*b**3*d*e**2/5 + 12* b**2*c*d**2*e/5 + b*c**2*d**3) + x**4*(a**2*b*e**3/4 + 3*a**2*c*d*e**2/2 + 3*a*b**2*d*e**2/2 + 9*a*b*c*d**2*e/2 + a*c**2*d**3 + 3*b**3*d**2*e/4 + b* *2*c*d**3) + x**3*(a**2*b*d*e**2 + 2*a**2*c*d**2*e + 2*a*b**2*d**2*e + 2*a *b*c*d**3 + b**3*d**3/3) + x**2*(3*a**2*b*d**2*e/2 + a**2*c*d**3 + a*b**2* d**3)
Time = 0.18 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.43 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{9} \, c^{3} e^{3} x^{9} + \frac {1}{8} \, {\left (6 \, c^{3} d e^{2} + 5 \, b c^{2} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, c^{3} d^{2} e + 15 \, b c^{2} d e^{2} + 4 \, {\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{7} + a^{2} b d^{3} x + \frac {1}{6} \, {\left (2 \, c^{3} d^{3} + 15 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} + {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, b c^{2} d^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e + 3 \, {\left (b^{3} + 6 \, a b c\right )} d e^{2} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (a^{2} b e^{3} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} + 3 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e + 6 \, {\left (a b^{2} + a^{2} c\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a^{2} b d e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{3} + 6 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{2} e + 2 \, {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} x^{2} \]
2/9*c^3*e^3*x^9 + 1/8*(6*c^3*d*e^2 + 5*b*c^2*e^3)*x^8 + 1/7*(6*c^3*d^2*e + 15*b*c^2*d*e^2 + 4*(b^2*c + a*c^2)*e^3)*x^7 + a^2*b*d^3*x + 1/6*(2*c^3*d^ 3 + 15*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 + (b^3 + 6*a*b*c)*e^3)*x^6 + 1/5*(5*b*c^2*d^3 + 12*(b^2*c + a*c^2)*d^2*e + 3*(b^3 + 6*a*b*c)*d*e^2 + 2 *(a*b^2 + a^2*c)*e^3)*x^5 + 1/4*(a^2*b*e^3 + 4*(b^2*c + a*c^2)*d^3 + 3*(b^ 3 + 6*a*b*c)*d^2*e + 6*(a*b^2 + a^2*c)*d*e^2)*x^4 + 1/3*(3*a^2*b*d*e^2 + ( b^3 + 6*a*b*c)*d^3 + 6*(a*b^2 + a^2*c)*d^2*e)*x^3 + 1/2*(3*a^2*b*d^2*e + 2 *(a*b^2 + a^2*c)*d^3)*x^2
Time = 0.26 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.79 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=\frac {2}{9} \, c^{3} e^{3} x^{9} + \frac {3}{4} \, c^{3} d e^{2} x^{8} + \frac {5}{8} \, b c^{2} e^{3} x^{8} + \frac {6}{7} \, c^{3} d^{2} e x^{7} + \frac {15}{7} \, b c^{2} d e^{2} x^{7} + \frac {4}{7} \, b^{2} c e^{3} x^{7} + \frac {4}{7} \, a c^{2} e^{3} x^{7} + \frac {1}{3} \, c^{3} d^{3} x^{6} + \frac {5}{2} \, b c^{2} d^{2} e x^{6} + 2 \, b^{2} c d e^{2} x^{6} + 2 \, a c^{2} d e^{2} x^{6} + \frac {1}{6} \, b^{3} e^{3} x^{6} + a b c e^{3} x^{6} + b c^{2} d^{3} x^{5} + \frac {12}{5} \, b^{2} c d^{2} e x^{5} + \frac {12}{5} \, a c^{2} d^{2} e x^{5} + \frac {3}{5} \, b^{3} d e^{2} x^{5} + \frac {18}{5} \, a b c d e^{2} x^{5} + \frac {2}{5} \, a b^{2} e^{3} x^{5} + \frac {2}{5} \, a^{2} c e^{3} x^{5} + b^{2} c d^{3} x^{4} + a c^{2} d^{3} x^{4} + \frac {3}{4} \, b^{3} d^{2} e x^{4} + \frac {9}{2} \, a b c d^{2} e x^{4} + \frac {3}{2} \, a b^{2} d e^{2} x^{4} + \frac {3}{2} \, a^{2} c d e^{2} x^{4} + \frac {1}{4} \, a^{2} b e^{3} x^{4} + \frac {1}{3} \, b^{3} d^{3} x^{3} + 2 \, a b c d^{3} x^{3} + 2 \, a b^{2} d^{2} e x^{3} + 2 \, a^{2} c d^{2} e x^{3} + a^{2} b d e^{2} x^{3} + a b^{2} d^{3} x^{2} + a^{2} c d^{3} x^{2} + \frac {3}{2} \, a^{2} b d^{2} e x^{2} + a^{2} b d^{3} x \]
2/9*c^3*e^3*x^9 + 3/4*c^3*d*e^2*x^8 + 5/8*b*c^2*e^3*x^8 + 6/7*c^3*d^2*e*x^ 7 + 15/7*b*c^2*d*e^2*x^7 + 4/7*b^2*c*e^3*x^7 + 4/7*a*c^2*e^3*x^7 + 1/3*c^3 *d^3*x^6 + 5/2*b*c^2*d^2*e*x^6 + 2*b^2*c*d*e^2*x^6 + 2*a*c^2*d*e^2*x^6 + 1 /6*b^3*e^3*x^6 + a*b*c*e^3*x^6 + b*c^2*d^3*x^5 + 12/5*b^2*c*d^2*e*x^5 + 12 /5*a*c^2*d^2*e*x^5 + 3/5*b^3*d*e^2*x^5 + 18/5*a*b*c*d*e^2*x^5 + 2/5*a*b^2* e^3*x^5 + 2/5*a^2*c*e^3*x^5 + b^2*c*d^3*x^4 + a*c^2*d^3*x^4 + 3/4*b^3*d^2* e*x^4 + 9/2*a*b*c*d^2*e*x^4 + 3/2*a*b^2*d*e^2*x^4 + 3/2*a^2*c*d*e^2*x^4 + 1/4*a^2*b*e^3*x^4 + 1/3*b^3*d^3*x^3 + 2*a*b*c*d^3*x^3 + 2*a*b^2*d^2*e*x^3 + 2*a^2*c*d^2*e*x^3 + a^2*b*d*e^2*x^3 + a*b^2*d^3*x^2 + a^2*c*d^3*x^2 + 3/ 2*a^2*b*d^2*e*x^2 + a^2*b*d^3*x
Time = 0.11 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.45 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=x^6\,\left (\frac {b^3\,e^3}{6}+2\,b^2\,c\,d\,e^2+\frac {5\,b\,c^2\,d^2\,e}{2}+a\,b\,c\,e^3+\frac {c^3\,d^3}{3}+2\,a\,c^2\,d\,e^2\right )+x^4\,\left (\frac {a^2\,b\,e^3}{4}+\frac {3\,a^2\,c\,d\,e^2}{2}+\frac {3\,a\,b^2\,d\,e^2}{2}+\frac {9\,a\,b\,c\,d^2\,e}{2}+a\,c^2\,d^3+\frac {3\,b^3\,d^2\,e}{4}+b^2\,c\,d^3\right )+x^5\,\left (\frac {2\,a^2\,c\,e^3}{5}+\frac {2\,a\,b^2\,e^3}{5}+\frac {18\,a\,b\,c\,d\,e^2}{5}+\frac {12\,a\,c^2\,d^2\,e}{5}+\frac {3\,b^3\,d\,e^2}{5}+\frac {12\,b^2\,c\,d^2\,e}{5}+b\,c^2\,d^3\right )+x^3\,\left (a^2\,b\,d\,e^2+2\,c\,a^2\,d^2\,e+2\,a\,b^2\,d^2\,e+2\,c\,a\,b\,d^3+\frac {b^3\,d^3}{3}\right )+\frac {2\,c^3\,e^3\,x^9}{9}+\frac {a\,d^2\,x^2\,\left (2\,d\,b^2+3\,a\,e\,b+2\,a\,c\,d\right )}{2}+\frac {c^2\,e^2\,x^8\,\left (5\,b\,e+6\,c\,d\right )}{8}+a^2\,b\,d^3\,x+\frac {c\,e\,x^7\,\left (4\,b^2\,e^2+15\,b\,c\,d\,e+6\,c^2\,d^2+4\,a\,c\,e^2\right )}{7} \]
x^6*((b^3*e^3)/6 + (c^3*d^3)/3 + a*b*c*e^3 + 2*a*c^2*d*e^2 + (5*b*c^2*d^2* e)/2 + 2*b^2*c*d*e^2) + x^4*(a*c^2*d^3 + (a^2*b*e^3)/4 + b^2*c*d^3 + (3*b^ 3*d^2*e)/4 + (3*a*b^2*d*e^2)/2 + (3*a^2*c*d*e^2)/2 + (9*a*b*c*d^2*e)/2) + x^5*((2*a*b^2*e^3)/5 + b*c^2*d^3 + (2*a^2*c*e^3)/5 + (3*b^3*d*e^2)/5 + (12 *a*c^2*d^2*e)/5 + (12*b^2*c*d^2*e)/5 + (18*a*b*c*d*e^2)/5) + x^3*((b^3*d^3 )/3 + 2*a*b*c*d^3 + 2*a*b^2*d^2*e + a^2*b*d*e^2 + 2*a^2*c*d^2*e) + (2*c^3* e^3*x^9)/9 + (a*d^2*x^2*(2*b^2*d + 3*a*b*e + 2*a*c*d))/2 + (c^2*e^2*x^8*(5 *b*e + 6*c*d))/8 + a^2*b*d^3*x + (c*e*x^7*(4*b^2*e^2 + 6*c^2*d^2 + 4*a*c*e ^2 + 15*b*c*d*e))/7