Integrand size = 18, antiderivative size = 158 \[ \int (A+B x) \left (a+b x+c x^2\right )^3 \, dx=a^3 A x+\frac {1}{2} a^2 (3 A b+a B) x^2+a \left (a b B+A \left (b^2+a c\right )\right ) x^3+\frac {1}{4} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^4+\frac {1}{5} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^5+\frac {1}{2} c \left (b^2 B+A b c+a B c\right ) x^6+\frac {1}{7} c^2 (3 b B+A c) x^7+\frac {1}{8} B c^3 x^8 \]
a^3*A*x+1/2*a^2*(3*A*b+B*a)*x^2+a*(a*b*B+A*(a*c+b^2))*x^3+1/4*(3*a*B*(a*c+ b^2)+A*(6*a*b*c+b^3))*x^4+1/5*(3*A*a*c^2+3*A*b^2*c+6*B*a*b*c+B*b^3)*x^5+1/ 2*c*(A*b*c+B*a*c+B*b^2)*x^6+1/7*c^2*(A*c+3*B*b)*x^7+1/8*B*c^3*x^8
Time = 0.02 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00 \[ \int (A+B x) \left (a+b x+c x^2\right )^3 \, dx=a^3 A x+\frac {1}{2} a^2 (3 A b+a B) x^2+a \left (a b B+A \left (b^2+a c\right )\right ) x^3+\frac {1}{4} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^4+\frac {1}{5} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^5+\frac {1}{2} c \left (b^2 B+A b c+a B c\right ) x^6+\frac {1}{7} c^2 (3 b B+A c) x^7+\frac {1}{8} B c^3 x^8 \]
a^3*A*x + (a^2*(3*A*b + a*B)*x^2)/2 + a*(a*b*B + A*(b^2 + a*c))*x^3 + ((3* a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^4)/4 + ((b^3*B + 3*A*b^2*c + 6*a*b* B*c + 3*a*A*c^2)*x^5)/5 + (c*(b^2*B + A*b*c + a*B*c)*x^6)/2 + (c^2*(3*b*B + A*c)*x^7)/7 + (B*c^3*x^8)/8
Time = 0.38 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1140, 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.
| \(\displaystyle \int (A+B x) \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (a^3 A+a^2 x (a B+3 A b)+3 c x^5 \left (a B c+A b c+b^2 B\right )+3 a x^2 \left (A \left (a c+b^2\right )+a b B\right )+x^4 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+x^3 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+c^2 x^6 (A c+3 b B)+B c^3 x^7\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^3 A x+\frac {1}{2} a^2 x^2 (a B+3 A b)+\frac {1}{2} c x^6 \left (a B c+A b c+b^2 B\right )+a x^3 \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{5} x^5 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {1}{4} x^4 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{7} c^2 x^7 (A c+3 b B)+\frac {1}{8} B c^3 x^8\) |
a^3*A*x + (a^2*(3*A*b + a*B)*x^2)/2 + a*(a*b*B + A*(b^2 + a*c))*x^3 + ((3* a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^4)/4 + ((b^3*B + 3*A*b^2*c + 6*a*b* B*c + 3*a*A*c^2)*x^5)/5 + (c*(b^2*B + A*b*c + a*B*c)*x^6)/2 + (c^2*(3*b*B + A*c)*x^7)/7 + (B*c^3*x^8)/8
3.9.69.3.1 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*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 0.16 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.04
| method | result | size |
| norman | \(\frac {B \,c^{3} x^{8}}{8}+\left (\frac {1}{7} A \,c^{3}+\frac {3}{7} B b \,c^{2}\right ) x^{7}+\left (\frac {1}{2} A b \,c^{2}+\frac {1}{2} B a \,c^{2}+\frac {1}{2} B \,b^{2} c \right ) x^{6}+\left (\frac {3}{5} A a \,c^{2}+\frac {3}{5} A \,b^{2} c +\frac {6}{5} B a b c +\frac {1}{5} B \,b^{3}\right ) x^{5}+\left (\frac {3}{2} A a b c +\frac {1}{4} A \,b^{3}+\frac {3}{4} B \,a^{2} c +\frac {3}{4} B a \,b^{2}\right ) x^{4}+\left (A \,a^{2} c +A a \,b^{2}+B b \,a^{2}\right ) x^{3}+\left (\frac {3}{2} A \,a^{2} b +\frac {1}{2} B \,a^{3}\right ) x^{2}+a^{3} A x\) | \(164\) |
| gosper | \(\frac {1}{8} B \,c^{3} x^{8}+\frac {1}{7} A \,c^{3} x^{7}+\frac {3}{7} x^{7} B b \,c^{2}+\frac {1}{2} x^{6} A b \,c^{2}+\frac {1}{2} B a \,c^{2} x^{6}+\frac {1}{2} x^{6} B \,b^{2} c +\frac {3}{5} a A \,c^{2} x^{5}+\frac {3}{5} x^{5} A \,b^{2} c +\frac {6}{5} x^{5} B a b c +\frac {1}{5} B \,b^{3} x^{5}+\frac {3}{2} x^{4} A a b c +\frac {1}{4} A \,b^{3} x^{4}+\frac {3}{4} B \,a^{2} c \,x^{4}+\frac {3}{4} x^{4} B a \,b^{2}+a^{2} A c \,x^{3}+A a \,b^{2} x^{3}+B \,a^{2} b \,x^{3}+\frac {3}{2} x^{2} A \,a^{2} b +\frac {1}{2} B \,a^{3} x^{2}+a^{3} A x\) | \(188\) |
| risch | \(\frac {1}{8} B \,c^{3} x^{8}+\frac {1}{7} A \,c^{3} x^{7}+\frac {3}{7} x^{7} B b \,c^{2}+\frac {1}{2} x^{6} A b \,c^{2}+\frac {1}{2} B a \,c^{2} x^{6}+\frac {1}{2} x^{6} B \,b^{2} c +\frac {3}{5} a A \,c^{2} x^{5}+\frac {3}{5} x^{5} A \,b^{2} c +\frac {6}{5} x^{5} B a b c +\frac {1}{5} B \,b^{3} x^{5}+\frac {3}{2} x^{4} A a b c +\frac {1}{4} A \,b^{3} x^{4}+\frac {3}{4} B \,a^{2} c \,x^{4}+\frac {3}{4} x^{4} B a \,b^{2}+a^{2} A c \,x^{3}+A a \,b^{2} x^{3}+B \,a^{2} b \,x^{3}+\frac {3}{2} x^{2} A \,a^{2} b +\frac {1}{2} B \,a^{3} x^{2}+a^{3} A x\) | \(188\) |
| parallelrisch | \(\frac {1}{8} B \,c^{3} x^{8}+\frac {1}{7} A \,c^{3} x^{7}+\frac {3}{7} x^{7} B b \,c^{2}+\frac {1}{2} x^{6} A b \,c^{2}+\frac {1}{2} B a \,c^{2} x^{6}+\frac {1}{2} x^{6} B \,b^{2} c +\frac {3}{5} a A \,c^{2} x^{5}+\frac {3}{5} x^{5} A \,b^{2} c +\frac {6}{5} x^{5} B a b c +\frac {1}{5} B \,b^{3} x^{5}+\frac {3}{2} x^{4} A a b c +\frac {1}{4} A \,b^{3} x^{4}+\frac {3}{4} B \,a^{2} c \,x^{4}+\frac {3}{4} x^{4} B a \,b^{2}+a^{2} A c \,x^{3}+A a \,b^{2} x^{3}+B \,a^{2} b \,x^{3}+\frac {3}{2} x^{2} A \,a^{2} b +\frac {1}{2} B \,a^{3} x^{2}+a^{3} A x\) | \(188\) |
| default | \(\frac {B \,c^{3} x^{8}}{8}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{7}}{7}+\frac {\left (3 A b \,c^{2}+B \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (A \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+B \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{5}}{5}+\frac {\left (A \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+B \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )\right ) x^{4}}{4}+\frac {\left (A \left (a \left (2 a c +b^{2}\right )+2 b^{2} a +c \,a^{2}\right )+3 B b \,a^{2}\right ) x^{3}}{3}+\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) x^{2}}{2}+a^{3} A x\) | \(223\) |
1/8*B*c^3*x^8+(1/7*A*c^3+3/7*B*b*c^2)*x^7+(1/2*A*b*c^2+1/2*B*a*c^2+1/2*B*b ^2*c)*x^6+(3/5*A*a*c^2+3/5*A*b^2*c+6/5*B*a*b*c+1/5*B*b^3)*x^5+(3/2*A*a*b*c +1/4*A*b^3+3/4*B*a^2*c+3/4*B*a*b^2)*x^4+(A*a^2*c+A*a*b^2+B*a^2*b)*x^3+(3/2 *A*a^2*b+1/2*B*a^3)*x^2+a^3*A*x
Time = 0.31 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.03 \[ \int (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{8} \, B c^{3} x^{8} + \frac {1}{7} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{5} + A a^{3} x + \frac {1}{4} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{4} + {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \]
1/8*B*c^3*x^8 + 1/7*(3*B*b*c^2 + A*c^3)*x^7 + 1/2*(B*b^2*c + (B*a + A*b)*c ^2)*x^6 + 1/5*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^5 + A*a^3*x + 1/4*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^4 + (B*a^2*b + A*a*b^2 + A*a^2*c)*x^3 + 1/2*(B*a^3 + 3*A*a^2*b)*x^2
Time = 0.03 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.20 \[ \int (A+B x) \left (a+b x+c x^2\right )^3 \, dx=A a^{3} x + \frac {B c^{3} x^{8}}{8} + x^{7} \left (\frac {A c^{3}}{7} + \frac {3 B b c^{2}}{7}\right ) + x^{6} \left (\frac {A b c^{2}}{2} + \frac {B a c^{2}}{2} + \frac {B b^{2} c}{2}\right ) + x^{5} \cdot \left (\frac {3 A a c^{2}}{5} + \frac {3 A b^{2} c}{5} + \frac {6 B a b c}{5} + \frac {B b^{3}}{5}\right ) + x^{4} \cdot \left (\frac {3 A a b c}{2} + \frac {A b^{3}}{4} + \frac {3 B a^{2} c}{4} + \frac {3 B a b^{2}}{4}\right ) + x^{3} \left (A a^{2} c + A a b^{2} + B a^{2} b\right ) + x^{2} \cdot \left (\frac {3 A a^{2} b}{2} + \frac {B a^{3}}{2}\right ) \]
A*a**3*x + B*c**3*x**8/8 + x**7*(A*c**3/7 + 3*B*b*c**2/7) + x**6*(A*b*c**2 /2 + B*a*c**2/2 + B*b**2*c/2) + x**5*(3*A*a*c**2/5 + 3*A*b**2*c/5 + 6*B*a* b*c/5 + B*b**3/5) + x**4*(3*A*a*b*c/2 + A*b**3/4 + 3*B*a**2*c/4 + 3*B*a*b* *2/4) + x**3*(A*a**2*c + A*a*b**2 + B*a**2*b) + x**2*(3*A*a**2*b/2 + B*a** 3/2)
Time = 0.20 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.03 \[ \int (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{8} \, B c^{3} x^{8} + \frac {1}{7} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{5} + A a^{3} x + \frac {1}{4} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{4} + {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \]
1/8*B*c^3*x^8 + 1/7*(3*B*b*c^2 + A*c^3)*x^7 + 1/2*(B*b^2*c + (B*a + A*b)*c ^2)*x^6 + 1/5*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^5 + A*a^3*x + 1/4*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^4 + (B*a^2*b + A*a*b^2 + A*a^2*c)*x^3 + 1/2*(B*a^3 + 3*A*a^2*b)*x^2
Time = 0.28 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.18 \[ \int (A+B x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{8} \, B c^{3} x^{8} + \frac {3}{7} \, B b c^{2} x^{7} + \frac {1}{7} \, A c^{3} x^{7} + \frac {1}{2} \, B b^{2} c x^{6} + \frac {1}{2} \, B a c^{2} x^{6} + \frac {1}{2} \, A b c^{2} x^{6} + \frac {1}{5} \, B b^{3} x^{5} + \frac {6}{5} \, B a b c x^{5} + \frac {3}{5} \, A b^{2} c x^{5} + \frac {3}{5} \, A a c^{2} x^{5} + \frac {3}{4} \, B a b^{2} x^{4} + \frac {1}{4} \, A b^{3} x^{4} + \frac {3}{4} \, B a^{2} c x^{4} + \frac {3}{2} \, A a b c x^{4} + B a^{2} b x^{3} + A a b^{2} x^{3} + A a^{2} c x^{3} + \frac {1}{2} \, B a^{3} x^{2} + \frac {3}{2} \, A a^{2} b x^{2} + A a^{3} x \]
1/8*B*c^3*x^8 + 3/7*B*b*c^2*x^7 + 1/7*A*c^3*x^7 + 1/2*B*b^2*c*x^6 + 1/2*B* a*c^2*x^6 + 1/2*A*b*c^2*x^6 + 1/5*B*b^3*x^5 + 6/5*B*a*b*c*x^5 + 3/5*A*b^2* c*x^5 + 3/5*A*a*c^2*x^5 + 3/4*B*a*b^2*x^4 + 1/4*A*b^3*x^4 + 3/4*B*a^2*c*x^ 4 + 3/2*A*a*b*c*x^4 + B*a^2*b*x^3 + A*a*b^2*x^3 + A*a^2*c*x^3 + 1/2*B*a^3* x^2 + 3/2*A*a^2*b*x^2 + A*a^3*x
Time = 0.05 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.03 \[ \int (A+B x) \left (a+b x+c x^2\right )^3 \, dx=x^4\,\left (\frac {3\,B\,c\,a^2}{4}+\frac {3\,B\,a\,b^2}{4}+\frac {3\,A\,c\,a\,b}{2}+\frac {A\,b^3}{4}\right )+x^5\,\left (\frac {B\,b^3}{5}+\frac {3\,A\,b^2\,c}{5}+\frac {6\,B\,a\,b\,c}{5}+\frac {3\,A\,a\,c^2}{5}\right )+x^2\,\left (\frac {B\,a^3}{2}+\frac {3\,A\,b\,a^2}{2}\right )+x^7\,\left (\frac {A\,c^3}{7}+\frac {3\,B\,b\,c^2}{7}\right )+x^3\,\left (B\,a^2\,b+A\,c\,a^2+A\,a\,b^2\right )+x^6\,\left (\frac {B\,b^2\,c}{2}+\frac {A\,b\,c^2}{2}+\frac {B\,a\,c^2}{2}\right )+\frac {B\,c^3\,x^8}{8}+A\,a^3\,x \]
x^4*((A*b^3)/4 + (3*B*a*b^2)/4 + (3*B*a^2*c)/4 + (3*A*a*b*c)/2) + x^5*((B* b^3)/5 + (3*A*a*c^2)/5 + (3*A*b^2*c)/5 + (6*B*a*b*c)/5) + x^2*((B*a^3)/2 + (3*A*a^2*b)/2) + x^7*((A*c^3)/7 + (3*B*b*c^2)/7) + x^3*(A*a*b^2 + A*a^2*c + B*a^2*b) + x^6*((A*b*c^2)/2 + (B*a*c^2)/2 + (B*b^2*c)/2) + (B*c^3*x^8)/ 8 + A*a^3*x