Integrand size = 29, antiderivative size = 92 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=\frac {(b c-a d)^3 (a+b x)^7}{7 b^4}+\frac {3 d (b c-a d)^2 (a+b x)^8}{8 b^4}+\frac {d^2 (b c-a d) (a+b x)^9}{3 b^4}+\frac {d^3 (a+b x)^{10}}{10 b^4} \]
1/7*(-a*d+b*c)^3*(b*x+a)^7/b^4+3/8*d*(-a*d+b*c)^2*(b*x+a)^8/b^4+1/3*d^2*(- a*d+b*c)*(b*x+a)^9/b^4+1/10*d^3*(b*x+a)^10/b^4
Leaf count is larger than twice the leaf count of optimal. \(276\) vs. \(2(92)=184\).
Time = 0.05 (sec) , antiderivative size = 276, normalized size of antiderivative = 3.00 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=\frac {1}{840} x \left (210 a^6 \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+252 a^5 b x \left (10 c^3+20 c^2 d x+15 c d^2 x^2+4 d^3 x^3\right )+210 a^4 b^2 x^2 \left (20 c^3+45 c^2 d x+36 c d^2 x^2+10 d^3 x^3\right )+120 a^3 b^3 x^3 \left (35 c^3+84 c^2 d x+70 c d^2 x^2+20 d^3 x^3\right )+45 a^2 b^4 x^4 \left (56 c^3+140 c^2 d x+120 c d^2 x^2+35 d^3 x^3\right )+10 a b^5 x^5 \left (84 c^3+216 c^2 d x+189 c d^2 x^2+56 d^3 x^3\right )+b^6 x^6 \left (120 c^3+315 c^2 d x+280 c d^2 x^2+84 d^3 x^3\right )\right ) \]
(x*(210*a^6*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + 252*a^5*b*x*(10* c^3 + 20*c^2*d*x + 15*c*d^2*x^2 + 4*d^3*x^3) + 210*a^4*b^2*x^2*(20*c^3 + 4 5*c^2*d*x + 36*c*d^2*x^2 + 10*d^3*x^3) + 120*a^3*b^3*x^3*(35*c^3 + 84*c^2* d*x + 70*c*d^2*x^2 + 20*d^3*x^3) + 45*a^2*b^4*x^4*(56*c^3 + 140*c^2*d*x + 120*c*d^2*x^2 + 35*d^3*x^3) + 10*a*b^5*x^5*(84*c^3 + 216*c^2*d*x + 189*c*d ^2*x^2 + 56*d^3*x^3) + b^6*x^6*(120*c^3 + 315*c^2*d*x + 280*c*d^2*x^2 + 84 *d^3*x^3)))/840
Time = 0.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1121, 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.
| \(\displaystyle \int (a+b x)^3 \left (x (a d+b c)+a c+b d x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (\frac {3 d^2 (a+b x)^8 (b c-a d)}{b^3}+\frac {3 d (a+b x)^7 (b c-a d)^2}{b^3}+\frac {(a+b x)^6 (b c-a d)^3}{b^3}+\frac {d^3 (a+b x)^9}{b^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 (a+b x)^9 (b c-a d)}{3 b^4}+\frac {3 d (a+b x)^8 (b c-a d)^2}{8 b^4}+\frac {(a+b x)^7 (b c-a d)^3}{7 b^4}+\frac {d^3 (a+b x)^{10}}{10 b^4}\) |
((b*c - a*d)^3*(a + b*x)^7)/(7*b^4) + (3*d*(b*c - a*d)^2*(a + b*x)^8)/(8*b ^4) + (d^2*(b*c - a*d)*(a + b*x)^9)/(3*b^4) + (d^3*(a + b*x)^10)/(10*b^4)
3.18.83.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(324\) vs. \(2(84)=168\).
Time = 2.36 (sec) , antiderivative size = 325, normalized size of antiderivative = 3.53
| method | result | size |
| norman | \(\frac {d^{3} b^{6} x^{10}}{10}+\left (\frac {2}{3} d^{3} a \,b^{5}+\frac {1}{3} c \,d^{2} b^{6}\right ) x^{9}+\left (\frac {15}{8} d^{3} a^{2} b^{4}+\frac {9}{4} c \,d^{2} a \,b^{5}+\frac {3}{8} c^{2} d \,b^{6}\right ) x^{8}+\left (\frac {20}{7} a^{3} b^{3} d^{3}+\frac {45}{7} a^{2} b^{4} c \,d^{2}+\frac {18}{7} c^{2} d a \,b^{5}+\frac {1}{7} c^{3} b^{6}\right ) x^{7}+\left (\frac {5}{2} d^{3} a^{4} b^{2}+10 c \,d^{2} a^{3} b^{3}+\frac {15}{2} c^{2} d \,a^{2} b^{4}+c^{3} a \,b^{5}\right ) x^{6}+\left (\frac {6}{5} a^{5} b \,d^{3}+9 b^{2} c \,d^{2} a^{4}+12 a^{3} b^{3} c^{2} d +3 a^{2} b^{4} c^{3}\right ) x^{5}+\left (\frac {1}{4} d^{3} a^{6}+\frac {9}{2} c \,d^{2} a^{5} b +\frac {45}{4} c^{2} d \,a^{4} b^{2}+5 a^{3} b^{3} c^{3}\right ) x^{4}+\left (c \,d^{2} a^{6}+6 c^{2} d \,a^{5} b +5 c^{3} a^{4} b^{2}\right ) x^{3}+\left (\frac {3}{2} c^{2} d \,a^{6}+3 c^{3} a^{5} b \right ) x^{2}+c^{3} a^{6} x\) | \(325\) |
| risch | \(\frac {20}{7} x^{7} a^{3} b^{3} d^{3}+\frac {5}{2} x^{6} d^{3} a^{4} b^{2}+x^{6} c^{3} a \,b^{5}+\frac {1}{7} x^{7} c^{3} b^{6}+9 x^{5} b^{2} c \,d^{2} a^{4}+12 x^{5} a^{3} b^{3} c^{2} d +\frac {9}{2} x^{4} c \,d^{2} a^{5} b +\frac {45}{4} x^{4} c^{2} d \,a^{4} b^{2}+\frac {1}{4} x^{4} d^{3} a^{6}+\frac {1}{10} d^{3} b^{6} x^{10}+c^{3} a^{6} x +\frac {2}{3} x^{9} d^{3} a \,b^{5}+\frac {1}{3} x^{9} c \,d^{2} b^{6}+\frac {15}{8} x^{8} d^{3} a^{2} b^{4}+\frac {3}{8} x^{8} c^{2} d \,b^{6}+\frac {6}{5} x^{5} a^{5} b \,d^{3}+3 x^{5} a^{2} b^{4} c^{3}+5 x^{4} a^{3} b^{3} c^{3}+\frac {3}{2} x^{2} c^{2} d \,a^{6}+3 x^{2} c^{3} a^{5} b +a^{6} c \,d^{2} x^{3}+5 a^{4} b^{2} c^{3} x^{3}+6 a^{5} b \,c^{2} d \,x^{3}+\frac {9}{4} x^{8} c \,d^{2} a \,b^{5}+\frac {45}{7} x^{7} a^{2} b^{4} c \,d^{2}+\frac {18}{7} x^{7} c^{2} d a \,b^{5}+10 x^{6} c \,d^{2} a^{3} b^{3}+\frac {15}{2} x^{6} c^{2} d \,a^{2} b^{4}\) | \(363\) |
| parallelrisch | \(\frac {20}{7} x^{7} a^{3} b^{3} d^{3}+\frac {5}{2} x^{6} d^{3} a^{4} b^{2}+x^{6} c^{3} a \,b^{5}+\frac {1}{7} x^{7} c^{3} b^{6}+9 x^{5} b^{2} c \,d^{2} a^{4}+12 x^{5} a^{3} b^{3} c^{2} d +\frac {9}{2} x^{4} c \,d^{2} a^{5} b +\frac {45}{4} x^{4} c^{2} d \,a^{4} b^{2}+\frac {1}{4} x^{4} d^{3} a^{6}+\frac {1}{10} d^{3} b^{6} x^{10}+c^{3} a^{6} x +\frac {2}{3} x^{9} d^{3} a \,b^{5}+\frac {1}{3} x^{9} c \,d^{2} b^{6}+\frac {15}{8} x^{8} d^{3} a^{2} b^{4}+\frac {3}{8} x^{8} c^{2} d \,b^{6}+\frac {6}{5} x^{5} a^{5} b \,d^{3}+3 x^{5} a^{2} b^{4} c^{3}+5 x^{4} a^{3} b^{3} c^{3}+\frac {3}{2} x^{2} c^{2} d \,a^{6}+3 x^{2} c^{3} a^{5} b +a^{6} c \,d^{2} x^{3}+5 a^{4} b^{2} c^{3} x^{3}+6 a^{5} b \,c^{2} d \,x^{3}+\frac {9}{4} x^{8} c \,d^{2} a \,b^{5}+\frac {45}{7} x^{7} a^{2} b^{4} c \,d^{2}+\frac {18}{7} x^{7} c^{2} d a \,b^{5}+10 x^{6} c \,d^{2} a^{3} b^{3}+\frac {15}{2} x^{6} c^{2} d \,a^{2} b^{4}\) | \(363\) |
| gosper | \(\frac {x \left (84 d^{3} b^{6} x^{9}+560 x^{8} d^{3} a \,b^{5}+280 x^{8} c \,d^{2} b^{6}+1575 x^{7} d^{3} a^{2} b^{4}+1890 x^{7} c \,d^{2} a \,b^{5}+315 x^{7} c^{2} d \,b^{6}+2400 x^{6} a^{3} b^{3} d^{3}+5400 x^{6} a^{2} b^{4} c \,d^{2}+2160 x^{6} c^{2} d a \,b^{5}+120 x^{6} c^{3} b^{6}+2100 x^{5} d^{3} a^{4} b^{2}+8400 x^{5} c \,d^{2} a^{3} b^{3}+6300 x^{5} c^{2} d \,a^{2} b^{4}+840 x^{5} c^{3} a \,b^{5}+1008 x^{4} a^{5} b \,d^{3}+7560 x^{4} b^{2} c \,d^{2} a^{4}+10080 x^{4} a^{3} b^{3} c^{2} d +2520 x^{4} a^{2} b^{4} c^{3}+210 x^{3} d^{3} a^{6}+3780 x^{3} c \,d^{2} a^{5} b +9450 x^{3} c^{2} d \,a^{4} b^{2}+4200 x^{3} a^{3} b^{3} c^{3}+840 a^{6} c \,d^{2} x^{2}+5040 a^{5} b \,c^{2} d \,x^{2}+4200 a^{4} b^{2} c^{3} x^{2}+1260 x \,c^{2} d \,a^{6}+2520 x \,c^{3} a^{5} b +840 c^{3} a^{6}\right )}{840}\) | \(364\) |
| default | \(\frac {d^{3} b^{6} x^{10}}{10}+\frac {\left (3 d^{3} a \,b^{5}+3 b^{5} \left (a d +b c \right ) d^{2}\right ) x^{9}}{9}+\frac {\left (3 d^{3} a^{2} b^{4}+9 a \,b^{4} \left (a d +b c \right ) d^{2}+b^{3} \left (a \,b^{2} c \,d^{2}+2 \left (a d +b c \right )^{2} b d +b d \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right )\right ) x^{8}}{8}+\frac {\left (a^{3} b^{3} d^{3}+9 a^{2} b^{3} \left (a d +b c \right ) d^{2}+3 a \,b^{2} \left (a \,b^{2} c \,d^{2}+2 \left (a d +b c \right )^{2} b d +b d \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right )+b^{3} \left (4 a c b d \left (a d +b c \right )+\left (a d +b c \right ) \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right )\right ) x^{7}}{7}+\frac {\left (3 a^{3} \left (a d +b c \right ) b^{2} d^{2}+3 a^{2} b \left (a \,b^{2} c \,d^{2}+2 \left (a d +b c \right )^{2} b d +b d \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right )+3 a \,b^{2} \left (4 a c b d \left (a d +b c \right )+\left (a d +b c \right ) \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right )+b^{3} \left (a c \left (\left (a d +b c \right )^{2}+2 a b c d \right )+2 \left (a d +b c \right )^{2} a c +a^{2} b \,c^{2} d \right )\right ) x^{6}}{6}+\frac {\left (a^{3} \left (a \,b^{2} c \,d^{2}+2 \left (a d +b c \right )^{2} b d +b d \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right )+3 a^{2} b \left (4 a c b d \left (a d +b c \right )+\left (a d +b c \right ) \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right )+3 a \,b^{2} \left (a c \left (\left (a d +b c \right )^{2}+2 a b c d \right )+2 \left (a d +b c \right )^{2} a c +a^{2} b \,c^{2} d \right )+3 b^{3} a^{2} c^{2} \left (a d +b c \right )\right ) x^{5}}{5}+\frac {\left (a^{3} \left (4 a c b d \left (a d +b c \right )+\left (a d +b c \right ) \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right )+3 a^{2} b \left (a c \left (\left (a d +b c \right )^{2}+2 a b c d \right )+2 \left (a d +b c \right )^{2} a c +a^{2} b \,c^{2} d \right )+9 a^{3} b^{2} c^{2} \left (a d +b c \right )+a^{3} b^{3} c^{3}\right ) x^{4}}{4}+\frac {\left (a^{3} \left (a c \left (\left (a d +b c \right )^{2}+2 a b c d \right )+2 \left (a d +b c \right )^{2} a c +a^{2} b \,c^{2} d \right )+9 a^{4} b \,c^{2} \left (a d +b c \right )+3 c^{3} a^{4} b^{2}\right ) x^{3}}{3}+\frac {\left (3 a^{5} c^{2} \left (a d +b c \right )+3 c^{3} a^{5} b \right ) x^{2}}{2}+c^{3} a^{6} x\) | \(811\) |
1/10*d^3*b^6*x^10+(2/3*d^3*a*b^5+1/3*c*d^2*b^6)*x^9+(15/8*d^3*a^2*b^4+9/4* c*d^2*a*b^5+3/8*c^2*d*b^6)*x^8+(20/7*a^3*b^3*d^3+45/7*a^2*b^4*c*d^2+18/7*c ^2*d*a*b^5+1/7*c^3*b^6)*x^7+(5/2*d^3*a^4*b^2+10*c*d^2*a^3*b^3+15/2*c^2*d*a ^2*b^4+c^3*a*b^5)*x^6+(6/5*a^5*b*d^3+9*b^2*c*d^2*a^4+12*a^3*b^3*c^2*d+3*a^ 2*b^4*c^3)*x^5+(1/4*d^3*a^6+9/2*c*d^2*a^5*b+45/4*c^2*d*a^4*b^2+5*a^3*b^3*c ^3)*x^4+(a^6*c*d^2+6*a^5*b*c^2*d+5*a^4*b^2*c^3)*x^3+(3/2*c^2*d*a^6+3*c^3*a ^5*b)*x^2+c^3*a^6*x
Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (84) = 168\).
Time = 0.28 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.55 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=\frac {1}{10} \, b^{6} d^{3} x^{10} + a^{6} c^{3} x + \frac {1}{3} \, {\left (b^{6} c d^{2} + 2 \, a b^{5} d^{3}\right )} x^{9} + \frac {3}{8} \, {\left (b^{6} c^{2} d + 6 \, a b^{5} c d^{2} + 5 \, a^{2} b^{4} d^{3}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} c^{3} + 18 \, a b^{5} c^{2} d + 45 \, a^{2} b^{4} c d^{2} + 20 \, a^{3} b^{3} d^{3}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} c^{3} + 15 \, a^{2} b^{4} c^{2} d + 20 \, a^{3} b^{3} c d^{2} + 5 \, a^{4} b^{2} d^{3}\right )} x^{6} + \frac {3}{5} \, {\left (5 \, a^{2} b^{4} c^{3} + 20 \, a^{3} b^{3} c^{2} d + 15 \, a^{4} b^{2} c d^{2} + 2 \, a^{5} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} c^{3} + 45 \, a^{4} b^{2} c^{2} d + 18 \, a^{5} b c d^{2} + a^{6} d^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} c^{3} + 6 \, a^{5} b c^{2} d + a^{6} c d^{2}\right )} x^{3} + \frac {3}{2} \, {\left (2 \, a^{5} b c^{3} + a^{6} c^{2} d\right )} x^{2} \]
1/10*b^6*d^3*x^10 + a^6*c^3*x + 1/3*(b^6*c*d^2 + 2*a*b^5*d^3)*x^9 + 3/8*(b ^6*c^2*d + 6*a*b^5*c*d^2 + 5*a^2*b^4*d^3)*x^8 + 1/7*(b^6*c^3 + 18*a*b^5*c^ 2*d + 45*a^2*b^4*c*d^2 + 20*a^3*b^3*d^3)*x^7 + 1/2*(2*a*b^5*c^3 + 15*a^2*b ^4*c^2*d + 20*a^3*b^3*c*d^2 + 5*a^4*b^2*d^3)*x^6 + 3/5*(5*a^2*b^4*c^3 + 20 *a^3*b^3*c^2*d + 15*a^4*b^2*c*d^2 + 2*a^5*b*d^3)*x^5 + 1/4*(20*a^3*b^3*c^3 + 45*a^4*b^2*c^2*d + 18*a^5*b*c*d^2 + a^6*d^3)*x^4 + (5*a^4*b^2*c^3 + 6*a ^5*b*c^2*d + a^6*c*d^2)*x^3 + 3/2*(2*a^5*b*c^3 + a^6*c^2*d)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (80) = 160\).
Time = 0.06 (sec) , antiderivative size = 364, normalized size of antiderivative = 3.96 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=a^{6} c^{3} x + \frac {b^{6} d^{3} x^{10}}{10} + x^{9} \cdot \left (\frac {2 a b^{5} d^{3}}{3} + \frac {b^{6} c d^{2}}{3}\right ) + x^{8} \cdot \left (\frac {15 a^{2} b^{4} d^{3}}{8} + \frac {9 a b^{5} c d^{2}}{4} + \frac {3 b^{6} c^{2} d}{8}\right ) + x^{7} \cdot \left (\frac {20 a^{3} b^{3} d^{3}}{7} + \frac {45 a^{2} b^{4} c d^{2}}{7} + \frac {18 a b^{5} c^{2} d}{7} + \frac {b^{6} c^{3}}{7}\right ) + x^{6} \cdot \left (\frac {5 a^{4} b^{2} d^{3}}{2} + 10 a^{3} b^{3} c d^{2} + \frac {15 a^{2} b^{4} c^{2} d}{2} + a b^{5} c^{3}\right ) + x^{5} \cdot \left (\frac {6 a^{5} b d^{3}}{5} + 9 a^{4} b^{2} c d^{2} + 12 a^{3} b^{3} c^{2} d + 3 a^{2} b^{4} c^{3}\right ) + x^{4} \left (\frac {a^{6} d^{3}}{4} + \frac {9 a^{5} b c d^{2}}{2} + \frac {45 a^{4} b^{2} c^{2} d}{4} + 5 a^{3} b^{3} c^{3}\right ) + x^{3} \left (a^{6} c d^{2} + 6 a^{5} b c^{2} d + 5 a^{4} b^{2} c^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{6} c^{2} d}{2} + 3 a^{5} b c^{3}\right ) \]
a**6*c**3*x + b**6*d**3*x**10/10 + x**9*(2*a*b**5*d**3/3 + b**6*c*d**2/3) + x**8*(15*a**2*b**4*d**3/8 + 9*a*b**5*c*d**2/4 + 3*b**6*c**2*d/8) + x**7* (20*a**3*b**3*d**3/7 + 45*a**2*b**4*c*d**2/7 + 18*a*b**5*c**2*d/7 + b**6*c **3/7) + x**6*(5*a**4*b**2*d**3/2 + 10*a**3*b**3*c*d**2 + 15*a**2*b**4*c** 2*d/2 + a*b**5*c**3) + x**5*(6*a**5*b*d**3/5 + 9*a**4*b**2*c*d**2 + 12*a** 3*b**3*c**2*d + 3*a**2*b**4*c**3) + x**4*(a**6*d**3/4 + 9*a**5*b*c*d**2/2 + 45*a**4*b**2*c**2*d/4 + 5*a**3*b**3*c**3) + x**3*(a**6*c*d**2 + 6*a**5*b *c**2*d + 5*a**4*b**2*c**3) + x**2*(3*a**6*c**2*d/2 + 3*a**5*b*c**3)
Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (84) = 168\).
Time = 0.23 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.55 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=\frac {1}{10} \, b^{6} d^{3} x^{10} + a^{6} c^{3} x + \frac {1}{3} \, {\left (b^{6} c d^{2} + 2 \, a b^{5} d^{3}\right )} x^{9} + \frac {3}{8} \, {\left (b^{6} c^{2} d + 6 \, a b^{5} c d^{2} + 5 \, a^{2} b^{4} d^{3}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} c^{3} + 18 \, a b^{5} c^{2} d + 45 \, a^{2} b^{4} c d^{2} + 20 \, a^{3} b^{3} d^{3}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} c^{3} + 15 \, a^{2} b^{4} c^{2} d + 20 \, a^{3} b^{3} c d^{2} + 5 \, a^{4} b^{2} d^{3}\right )} x^{6} + \frac {3}{5} \, {\left (5 \, a^{2} b^{4} c^{3} + 20 \, a^{3} b^{3} c^{2} d + 15 \, a^{4} b^{2} c d^{2} + 2 \, a^{5} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} c^{3} + 45 \, a^{4} b^{2} c^{2} d + 18 \, a^{5} b c d^{2} + a^{6} d^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} c^{3} + 6 \, a^{5} b c^{2} d + a^{6} c d^{2}\right )} x^{3} + \frac {3}{2} \, {\left (2 \, a^{5} b c^{3} + a^{6} c^{2} d\right )} x^{2} \]
1/10*b^6*d^3*x^10 + a^6*c^3*x + 1/3*(b^6*c*d^2 + 2*a*b^5*d^3)*x^9 + 3/8*(b ^6*c^2*d + 6*a*b^5*c*d^2 + 5*a^2*b^4*d^3)*x^8 + 1/7*(b^6*c^3 + 18*a*b^5*c^ 2*d + 45*a^2*b^4*c*d^2 + 20*a^3*b^3*d^3)*x^7 + 1/2*(2*a*b^5*c^3 + 15*a^2*b ^4*c^2*d + 20*a^3*b^3*c*d^2 + 5*a^4*b^2*d^3)*x^6 + 3/5*(5*a^2*b^4*c^3 + 20 *a^3*b^3*c^2*d + 15*a^4*b^2*c*d^2 + 2*a^5*b*d^3)*x^5 + 1/4*(20*a^3*b^3*c^3 + 45*a^4*b^2*c^2*d + 18*a^5*b*c*d^2 + a^6*d^3)*x^4 + (5*a^4*b^2*c^3 + 6*a ^5*b*c^2*d + a^6*c*d^2)*x^3 + 3/2*(2*a^5*b*c^3 + a^6*c^2*d)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (84) = 168\).
Time = 0.29 (sec) , antiderivative size = 362, normalized size of antiderivative = 3.93 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=\frac {1}{10} \, b^{6} d^{3} x^{10} + \frac {1}{3} \, b^{6} c d^{2} x^{9} + \frac {2}{3} \, a b^{5} d^{3} x^{9} + \frac {3}{8} \, b^{6} c^{2} d x^{8} + \frac {9}{4} \, a b^{5} c d^{2} x^{8} + \frac {15}{8} \, a^{2} b^{4} d^{3} x^{8} + \frac {1}{7} \, b^{6} c^{3} x^{7} + \frac {18}{7} \, a b^{5} c^{2} d x^{7} + \frac {45}{7} \, a^{2} b^{4} c d^{2} x^{7} + \frac {20}{7} \, a^{3} b^{3} d^{3} x^{7} + a b^{5} c^{3} x^{6} + \frac {15}{2} \, a^{2} b^{4} c^{2} d x^{6} + 10 \, a^{3} b^{3} c d^{2} x^{6} + \frac {5}{2} \, a^{4} b^{2} d^{3} x^{6} + 3 \, a^{2} b^{4} c^{3} x^{5} + 12 \, a^{3} b^{3} c^{2} d x^{5} + 9 \, a^{4} b^{2} c d^{2} x^{5} + \frac {6}{5} \, a^{5} b d^{3} x^{5} + 5 \, a^{3} b^{3} c^{3} x^{4} + \frac {45}{4} \, a^{4} b^{2} c^{2} d x^{4} + \frac {9}{2} \, a^{5} b c d^{2} x^{4} + \frac {1}{4} \, a^{6} d^{3} x^{4} + 5 \, a^{4} b^{2} c^{3} x^{3} + 6 \, a^{5} b c^{2} d x^{3} + a^{6} c d^{2} x^{3} + 3 \, a^{5} b c^{3} x^{2} + \frac {3}{2} \, a^{6} c^{2} d x^{2} + a^{6} c^{3} x \]
1/10*b^6*d^3*x^10 + 1/3*b^6*c*d^2*x^9 + 2/3*a*b^5*d^3*x^9 + 3/8*b^6*c^2*d* x^8 + 9/4*a*b^5*c*d^2*x^8 + 15/8*a^2*b^4*d^3*x^8 + 1/7*b^6*c^3*x^7 + 18/7* a*b^5*c^2*d*x^7 + 45/7*a^2*b^4*c*d^2*x^7 + 20/7*a^3*b^3*d^3*x^7 + a*b^5*c^ 3*x^6 + 15/2*a^2*b^4*c^2*d*x^6 + 10*a^3*b^3*c*d^2*x^6 + 5/2*a^4*b^2*d^3*x^ 6 + 3*a^2*b^4*c^3*x^5 + 12*a^3*b^3*c^2*d*x^5 + 9*a^4*b^2*c*d^2*x^5 + 6/5*a ^5*b*d^3*x^5 + 5*a^3*b^3*c^3*x^4 + 45/4*a^4*b^2*c^2*d*x^4 + 9/2*a^5*b*c*d^ 2*x^4 + 1/4*a^6*d^3*x^4 + 5*a^4*b^2*c^3*x^3 + 6*a^5*b*c^2*d*x^3 + a^6*c*d^ 2*x^3 + 3*a^5*b*c^3*x^2 + 3/2*a^6*c^2*d*x^2 + a^6*c^3*x
Time = 9.86 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.35 \[ \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx=x^5\,\left (\frac {6\,a^5\,b\,d^3}{5}+9\,a^4\,b^2\,c\,d^2+12\,a^3\,b^3\,c^2\,d+3\,a^2\,b^4\,c^3\right )+x^6\,\left (\frac {5\,a^4\,b^2\,d^3}{2}+10\,a^3\,b^3\,c\,d^2+\frac {15\,a^2\,b^4\,c^2\,d}{2}+a\,b^5\,c^3\right )+x^4\,\left (\frac {a^6\,d^3}{4}+\frac {9\,a^5\,b\,c\,d^2}{2}+\frac {45\,a^4\,b^2\,c^2\,d}{4}+5\,a^3\,b^3\,c^3\right )+x^7\,\left (\frac {20\,a^3\,b^3\,d^3}{7}+\frac {45\,a^2\,b^4\,c\,d^2}{7}+\frac {18\,a\,b^5\,c^2\,d}{7}+\frac {b^6\,c^3}{7}\right )+a^6\,c^3\,x+\frac {b^6\,d^3\,x^{10}}{10}+\frac {3\,a^5\,c^2\,x^2\,\left (a\,d+2\,b\,c\right )}{2}+\frac {b^5\,d^2\,x^9\,\left (2\,a\,d+b\,c\right )}{3}+a^4\,c\,x^3\,\left (a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )+\frac {3\,b^4\,d\,x^8\,\left (5\,a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{8} \]
x^5*((6*a^5*b*d^3)/5 + 3*a^2*b^4*c^3 + 12*a^3*b^3*c^2*d + 9*a^4*b^2*c*d^2) + x^6*(a*b^5*c^3 + (5*a^4*b^2*d^3)/2 + (15*a^2*b^4*c^2*d)/2 + 10*a^3*b^3* c*d^2) + x^4*((a^6*d^3)/4 + 5*a^3*b^3*c^3 + (45*a^4*b^2*c^2*d)/4 + (9*a^5* b*c*d^2)/2) + x^7*((b^6*c^3)/7 + (20*a^3*b^3*d^3)/7 + (45*a^2*b^4*c*d^2)/7 + (18*a*b^5*c^2*d)/7) + a^6*c^3*x + (b^6*d^3*x^10)/10 + (3*a^5*c^2*x^2*(a *d + 2*b*c))/2 + (b^5*d^2*x^9*(2*a*d + b*c))/3 + a^4*c*x^3*(a^2*d^2 + 5*b^ 2*c^2 + 6*a*b*c*d) + (3*b^4*d*x^8*(5*a^2*d^2 + b^2*c^2 + 6*a*b*c*d))/8