Integrand size = 19, antiderivative size = 167 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^3 \, dx=a^3 \text {b1} x+\frac {1}{2} a^2 (6 b \text {b1}+a \text {c1}) x^2+a \left (4 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^3+\frac {1}{4} \left (8 b^3 \text {b1}+12 a b \text {b1} c+12 a b^2 \text {c1}+3 a^2 c \text {c1}\right ) x^4+\frac {1}{5} \left (12 b^2 \text {b1} c+3 a \text {b1} c^2+8 b^3 \text {c1}+12 a b c \text {c1}\right ) x^5+\frac {1}{2} c \left (2 b \text {b1} c+4 b^2 \text {c1}+a c \text {c1}\right ) x^6+\frac {1}{7} c^2 (\text {b1} c+6 b \text {c1}) x^7+\frac {1}{8} c^3 \text {c1} x^8 \] Output:
a^3*b1*x+1/2*a^2*(a*c1+6*b*b1)*x^2+a*(2*a*b*c1+a*b1*c+4*b^2*b1)*x^3+1/4*(3 *a^2*c*c1+12*a*b^2*c1+12*a*b*b1*c+8*b^3*b1)*x^4+1/5*(12*a*b*c*c1+3*a*b1*c^ 2+8*b^3*c1+12*b^2*b1*c)*x^5+1/2*c*(a*c*c1+4*b^2*c1+2*b*b1*c)*x^6+1/7*c^2*( 6*b*c1+b1*c)*x^7+1/8*c^3*c1*x^8
Time = 0.02 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^3 \, dx=a^3 \text {b1} x+\frac {1}{2} a^2 (6 b \text {b1}+a \text {c1}) x^2+a \left (4 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^3+\frac {1}{4} \left (8 b^3 \text {b1}+12 a b \text {b1} c+12 a b^2 \text {c1}+3 a^2 c \text {c1}\right ) x^4+\frac {1}{5} \left (12 b^2 \text {b1} c+3 a \text {b1} c^2+8 b^3 \text {c1}+12 a b c \text {c1}\right ) x^5+\frac {1}{2} c \left (2 b \text {b1} c+4 b^2 \text {c1}+a c \text {c1}\right ) x^6+\frac {1}{7} c^2 (\text {b1} c+6 b \text {c1}) x^7+\frac {1}{8} c^3 \text {c1} x^8 \] Input:
Integrate[(b1 + c1*x)*(a + 2*b*x + c*x^2)^3,x]
Output:
a^3*b1*x + (a^2*(6*b*b1 + a*c1)*x^2)/2 + a*(4*b^2*b1 + a*b1*c + 2*a*b*c1)* x^3 + ((8*b^3*b1 + 12*a*b*b1*c + 12*a*b^2*c1 + 3*a^2*c*c1)*x^4)/4 + ((12*b ^2*b1*c + 3*a*b1*c^2 + 8*b^3*c1 + 12*a*b*c*c1)*x^5)/5 + (c*(2*b*b1*c + 4*b ^2*c1 + a*c*c1)*x^6)/2 + (c^2*(b1*c + 6*b*c1)*x^7)/7 + (c^3*c1*x^8)/8
Time = 0.37 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.
| \(\displaystyle \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (a^3 \text {b1}+x^3 \left (3 a^2 c \text {c1}+12 a b^2 \text {c1}+12 a b \text {b1} c+8 b^3 \text {b1}\right )+a^2 x (a \text {c1}+6 b \text {b1})+3 c x^5 \left (a c \text {c1}+4 b^2 \text {c1}+2 b \text {b1} c\right )+3 a x^2 \left (2 a b \text {c1}+a \text {b1} c+4 b^2 \text {b1}\right )+x^4 \left (12 a b c \text {c1}+3 a \text {b1} c^2+8 b^3 \text {c1}+12 b^2 \text {b1} c\right )+c^2 x^6 (6 b \text {c1}+\text {b1} c)+c^3 \text {c1} x^7\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^3 \text {b1} x+\frac {1}{4} x^4 \left (3 a^2 c \text {c1}+12 a b^2 \text {c1}+12 a b \text {b1} c+8 b^3 \text {b1}\right )+\frac {1}{2} a^2 x^2 (a \text {c1}+6 b \text {b1})+\frac {1}{2} c x^6 \left (a c \text {c1}+4 b^2 \text {c1}+2 b \text {b1} c\right )+a x^3 \left (2 a b \text {c1}+a \text {b1} c+4 b^2 \text {b1}\right )+\frac {1}{5} x^5 \left (12 a b c \text {c1}+3 a \text {b1} c^2+8 b^3 \text {c1}+12 b^2 \text {b1} c\right )+\frac {1}{7} c^2 x^7 (6 b \text {c1}+\text {b1} c)+\frac {1}{8} c^3 \text {c1} x^8\) |
Input:
Int[(b1 + c1*x)*(a + 2*b*x + c*x^2)^3,x]
Output:
a^3*b1*x + (a^2*(6*b*b1 + a*c1)*x^2)/2 + a*(4*b^2*b1 + a*b1*c + 2*a*b*c1)* x^3 + ((8*b^3*b1 + 12*a*b*b1*c + 12*a*b^2*c1 + 3*a^2*c*c1)*x^4)/4 + ((12*b ^2*b1*c + 3*a*b1*c^2 + 8*b^3*c1 + 12*a*b*c*c1)*x^5)/5 + (c*(2*b*b1*c + 4*b ^2*c1 + a*c*c1)*x^6)/2 + (c^2*(b1*c + 6*b*c1)*x^7)/7 + (c^3*c1*x^8)/8
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 = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.99
| method | result | size |
| norman | \(\frac {c^{3} \operatorname {c1} \,x^{8}}{8}+\left (\frac {6}{7} \operatorname {c1} b \,c^{2}+\frac {1}{7} \operatorname {b1} \,c^{3}\right ) x^{7}+\left (\frac {1}{2} a \,c^{2} \operatorname {c1} +2 b^{2} c \operatorname {c1} +\operatorname {b1} b \,c^{2}\right ) x^{6}+\left (\frac {12}{5} a b c \operatorname {c1} +\frac {3}{5} a \operatorname {b1} \,c^{2}+\frac {8}{5} b^{3} \operatorname {c1} +\frac {12}{5} b^{2} \operatorname {b1} c \right ) x^{5}+\left (\frac {3}{4} a^{2} c \operatorname {c1} +3 a \,b^{2} \operatorname {c1} +3 a b \operatorname {b1} c +2 b^{3} \operatorname {b1} \right ) x^{4}+\left (2 \operatorname {c1} \,a^{2} b +a^{2} \operatorname {b1} c +4 a \,b^{2} \operatorname {b1} \right ) x^{3}+\left (\frac {1}{2} \operatorname {c1} \,a^{3}+3 \operatorname {b1} \,a^{2} b \right ) x^{2}+a^{3} \operatorname {b1} x\) | \(165\) |
| gosper | \(\frac {1}{8} c^{3} \operatorname {c1} \,x^{8}+\frac {6}{7} x^{7} \operatorname {c1} b \,c^{2}+\frac {1}{7} x^{7} \operatorname {b1} \,c^{3}+\frac {1}{2} x^{6} a \,c^{2} \operatorname {c1} +2 x^{6} b^{2} c \operatorname {c1} +x^{6} \operatorname {b1} b \,c^{2}+\frac {12}{5} x^{5} a b c \operatorname {c1} +\frac {3}{5} x^{5} a \operatorname {b1} \,c^{2}+\frac {8}{5} x^{5} b^{3} \operatorname {c1} +\frac {12}{5} x^{5} b^{2} \operatorname {b1} c +\frac {3}{4} x^{4} a^{2} c \operatorname {c1} +3 x^{4} a \,b^{2} \operatorname {c1} +3 x^{4} a b \operatorname {b1} c +2 x^{4} b^{3} \operatorname {b1} +2 a^{2} b \operatorname {c1} \,x^{3}+a^{2} \operatorname {b1} c \,x^{3}+4 a \,b^{2} \operatorname {b1} \,x^{3}+\frac {1}{2} x^{2} \operatorname {c1} \,a^{3}+3 x^{2} \operatorname {b1} \,a^{2} b +a^{3} \operatorname {b1} x\) | \(189\) |
| risch | \(\frac {1}{8} c^{3} \operatorname {c1} \,x^{8}+\frac {6}{7} x^{7} \operatorname {c1} b \,c^{2}+\frac {1}{7} x^{7} \operatorname {b1} \,c^{3}+\frac {1}{2} x^{6} a \,c^{2} \operatorname {c1} +2 x^{6} b^{2} c \operatorname {c1} +x^{6} \operatorname {b1} b \,c^{2}+\frac {12}{5} x^{5} a b c \operatorname {c1} +\frac {3}{5} x^{5} a \operatorname {b1} \,c^{2}+\frac {8}{5} x^{5} b^{3} \operatorname {c1} +\frac {12}{5} x^{5} b^{2} \operatorname {b1} c +\frac {3}{4} x^{4} a^{2} c \operatorname {c1} +3 x^{4} a \,b^{2} \operatorname {c1} +3 x^{4} a b \operatorname {b1} c +2 x^{4} b^{3} \operatorname {b1} +2 a^{2} b \operatorname {c1} \,x^{3}+a^{2} \operatorname {b1} c \,x^{3}+4 a \,b^{2} \operatorname {b1} \,x^{3}+\frac {1}{2} x^{2} \operatorname {c1} \,a^{3}+3 x^{2} \operatorname {b1} \,a^{2} b +a^{3} \operatorname {b1} x\) | \(189\) |
| parallelrisch | \(\frac {1}{8} c^{3} \operatorname {c1} \,x^{8}+\frac {6}{7} x^{7} \operatorname {c1} b \,c^{2}+\frac {1}{7} x^{7} \operatorname {b1} \,c^{3}+\frac {1}{2} x^{6} a \,c^{2} \operatorname {c1} +2 x^{6} b^{2} c \operatorname {c1} +x^{6} \operatorname {b1} b \,c^{2}+\frac {12}{5} x^{5} a b c \operatorname {c1} +\frac {3}{5} x^{5} a \operatorname {b1} \,c^{2}+\frac {8}{5} x^{5} b^{3} \operatorname {c1} +\frac {12}{5} x^{5} b^{2} \operatorname {b1} c +\frac {3}{4} x^{4} a^{2} c \operatorname {c1} +3 x^{4} a \,b^{2} \operatorname {c1} +3 x^{4} a b \operatorname {b1} c +2 x^{4} b^{3} \operatorname {b1} +2 a^{2} b \operatorname {c1} \,x^{3}+a^{2} \operatorname {b1} c \,x^{3}+4 a \,b^{2} \operatorname {b1} \,x^{3}+\frac {1}{2} x^{2} \operatorname {c1} \,a^{3}+3 x^{2} \operatorname {b1} \,a^{2} b +a^{3} \operatorname {b1} x\) | \(189\) |
| orering | \(\frac {x \left (35 \operatorname {c1} \,c^{3} x^{7}+240 b \,c^{2} \operatorname {c1} \,x^{6}+40 \operatorname {b1} \,c^{3} x^{6}+140 a \,c^{2} \operatorname {c1} \,x^{5}+560 b^{2} c \operatorname {c1} \,x^{5}+280 b \operatorname {b1} \,c^{2} x^{5}+672 a b c \operatorname {c1} \,x^{4}+168 a \operatorname {b1} \,c^{2} x^{4}+448 b^{3} \operatorname {c1} \,x^{4}+672 b^{2} \operatorname {b1} c \,x^{4}+210 a^{2} c \operatorname {c1} \,x^{3}+840 a \,b^{2} \operatorname {c1} \,x^{3}+840 a b \operatorname {b1} c \,x^{3}+560 b^{3} \operatorname {b1} \,x^{3}+560 a^{2} b \operatorname {c1} \,x^{2}+280 a^{2} \operatorname {b1} c \,x^{2}+1120 a \,b^{2} \operatorname {b1} \,x^{2}+140 a^{3} \operatorname {c1} x +840 a^{2} b \operatorname {b1} x +280 \operatorname {b1} \,a^{3}\right )}{280}\) | \(190\) |
| default | \(\frac {c^{3} \operatorname {c1} \,x^{8}}{8}+\frac {\left (6 \operatorname {c1} b \,c^{2}+\operatorname {b1} \,c^{3}\right ) x^{7}}{7}+\frac {\left (6 \operatorname {b1} b \,c^{2}+\operatorname {c1} \left (c^{2} a +8 b^{2} c +c \left (2 a c +4 b^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (\operatorname {b1} \left (c^{2} a +8 b^{2} c +c \left (2 a c +4 b^{2}\right )\right )+\operatorname {c1} \left (8 a b c +2 b \left (2 a c +4 b^{2}\right )\right )\right ) x^{5}}{5}+\frac {\left (\operatorname {b1} \left (8 a b c +2 b \left (2 a c +4 b^{2}\right )\right )+\operatorname {c1} \left (a \left (2 a c +4 b^{2}\right )+8 b^{2} a +c \,a^{2}\right )\right ) x^{4}}{4}+\frac {\left (\operatorname {b1} \left (a \left (2 a c +4 b^{2}\right )+8 b^{2} a +c \,a^{2}\right )+6 \operatorname {c1} \,a^{2} b \right ) x^{3}}{3}+\frac {\left (\operatorname {c1} \,a^{3}+6 \operatorname {b1} \,a^{2} b \right ) x^{2}}{2}+a^{3} \operatorname {b1} x\) | \(237\) |
Input:
int((c1*x+b1)*(c*x^2+2*b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/8*c^3*c1*x^8+(6/7*c1*b*c^2+1/7*b1*c^3)*x^7+(1/2*a*c^2*c1+2*b^2*c*c1+b1*b *c^2)*x^6+(12/5*a*b*c*c1+3/5*a*b1*c^2+8/5*b^3*c1+12/5*b^2*b1*c)*x^5+(3/4*a ^2*c*c1+3*a*b^2*c1+3*a*b*b1*c+2*b^3*b1)*x^4+(2*a^2*b*c1+a^2*b1*c+4*a*b^2*b 1)*x^3+(1/2*c1*a^3+3*b1*a^2*b)*x^2+a^3*b1*x
Time = 0.06 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.02 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} c_{1} x^{8} + \frac {1}{7} \, {\left (b_{1} c^{3} + 6 \, b c^{2} c_{1}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, b b_{1} c^{2} + {\left (4 \, b^{2} c + a c^{2}\right )} c_{1}\right )} x^{6} + \frac {1}{5} \, {\left (12 \, b^{2} b_{1} c + 3 \, a b_{1} c^{2} + 4 \, {\left (2 \, b^{3} + 3 \, a b c\right )} c_{1}\right )} x^{5} + a^{3} b_{1} x + \frac {1}{4} \, {\left (8 \, b^{3} b_{1} + 12 \, a b b_{1} c + 3 \, {\left (4 \, a b^{2} + a^{2} c\right )} c_{1}\right )} x^{4} + {\left (4 \, a b^{2} b_{1} + a^{2} b_{1} c + 2 \, a^{2} b c_{1}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{2} b b_{1} + a^{3} c_{1}\right )} x^{2} \] Input:
integrate((c1*x+b1)*(c*x^2+2*b*x+a)^3,x, algorithm="fricas")
Output:
1/8*c^3*c1*x^8 + 1/7*(b1*c^3 + 6*b*c^2*c1)*x^7 + 1/2*(2*b*b1*c^2 + (4*b^2* c + a*c^2)*c1)*x^6 + 1/5*(12*b^2*b1*c + 3*a*b1*c^2 + 4*(2*b^3 + 3*a*b*c)*c 1)*x^5 + a^3*b1*x + 1/4*(8*b^3*b1 + 12*a*b*b1*c + 3*(4*a*b^2 + a^2*c)*c1)* x^4 + (4*a*b^2*b1 + a^2*b1*c + 2*a^2*b*c1)*x^3 + 1/2*(6*a^2*b*b1 + a^3*c1) *x^2
Time = 0.03 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.13 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^3 \, dx=a^{3} b_{1} x + \frac {c^{3} c_{1} x^{8}}{8} + x^{7} \cdot \left (\frac {6 b c^{2} c_{1}}{7} + \frac {b_{1} c^{3}}{7}\right ) + x^{6} \left (\frac {a c^{2} c_{1}}{2} + 2 b^{2} c c_{1} + b b_{1} c^{2}\right ) + x^{5} \cdot \left (\frac {12 a b c c_{1}}{5} + \frac {3 a b_{1} c^{2}}{5} + \frac {8 b^{3} c_{1}}{5} + \frac {12 b^{2} b_{1} c}{5}\right ) + x^{4} \cdot \left (\frac {3 a^{2} c c_{1}}{4} + 3 a b^{2} c_{1} + 3 a b b_{1} c + 2 b^{3} b_{1}\right ) + x^{3} \cdot \left (2 a^{2} b c_{1} + a^{2} b_{1} c + 4 a b^{2} b_{1}\right ) + x^{2} \left (\frac {a^{3} c_{1}}{2} + 3 a^{2} b b_{1}\right ) \] Input:
integrate((c1*x+b1)*(c*x**2+2*b*x+a)**3,x)
Output:
a**3*b1*x + c**3*c1*x**8/8 + x**7*(6*b*c**2*c1/7 + b1*c**3/7) + x**6*(a*c* *2*c1/2 + 2*b**2*c*c1 + b*b1*c**2) + x**5*(12*a*b*c*c1/5 + 3*a*b1*c**2/5 + 8*b**3*c1/5 + 12*b**2*b1*c/5) + x**4*(3*a**2*c*c1/4 + 3*a*b**2*c1 + 3*a*b *b1*c + 2*b**3*b1) + x**3*(2*a**2*b*c1 + a**2*b1*c + 4*a*b**2*b1) + x**2*( a**3*c1/2 + 3*a**2*b*b1)
Time = 0.03 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.02 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} c_{1} x^{8} + \frac {1}{7} \, {\left (b_{1} c^{3} + 6 \, b c^{2} c_{1}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, b b_{1} c^{2} + {\left (4 \, b^{2} c + a c^{2}\right )} c_{1}\right )} x^{6} + \frac {1}{5} \, {\left (12 \, b^{2} b_{1} c + 3 \, a b_{1} c^{2} + 4 \, {\left (2 \, b^{3} + 3 \, a b c\right )} c_{1}\right )} x^{5} + a^{3} b_{1} x + \frac {1}{4} \, {\left (8 \, b^{3} b_{1} + 12 \, a b b_{1} c + 3 \, {\left (4 \, a b^{2} + a^{2} c\right )} c_{1}\right )} x^{4} + {\left (4 \, a b^{2} b_{1} + a^{2} b_{1} c + 2 \, a^{2} b c_{1}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{2} b b_{1} + a^{3} c_{1}\right )} x^{2} \] Input:
integrate((c1*x+b1)*(c*x^2+2*b*x+a)^3,x, algorithm="maxima")
Output:
1/8*c^3*c1*x^8 + 1/7*(b1*c^3 + 6*b*c^2*c1)*x^7 + 1/2*(2*b*b1*c^2 + (4*b^2* c + a*c^2)*c1)*x^6 + 1/5*(12*b^2*b1*c + 3*a*b1*c^2 + 4*(2*b^3 + 3*a*b*c)*c 1)*x^5 + a^3*b1*x + 1/4*(8*b^3*b1 + 12*a*b*b1*c + 3*(4*a*b^2 + a^2*c)*c1)* x^4 + (4*a*b^2*b1 + a^2*b1*c + 2*a^2*b*c1)*x^3 + 1/2*(6*a^2*b*b1 + a^3*c1) *x^2
Time = 0.12 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.13 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} c_{1} x^{8} + \frac {1}{7} \, b_{1} c^{3} x^{7} + \frac {6}{7} \, b c^{2} c_{1} x^{7} + b b_{1} c^{2} x^{6} + 2 \, b^{2} c c_{1} x^{6} + \frac {1}{2} \, a c^{2} c_{1} x^{6} + \frac {12}{5} \, b^{2} b_{1} c x^{5} + \frac {3}{5} \, a b_{1} c^{2} x^{5} + \frac {8}{5} \, b^{3} c_{1} x^{5} + \frac {12}{5} \, a b c c_{1} x^{5} + 2 \, b^{3} b_{1} x^{4} + 3 \, a b b_{1} c x^{4} + 3 \, a b^{2} c_{1} x^{4} + \frac {3}{4} \, a^{2} c c_{1} x^{4} + 4 \, a b^{2} b_{1} x^{3} + a^{2} b_{1} c x^{3} + 2 \, a^{2} b c_{1} x^{3} + 3 \, a^{2} b b_{1} x^{2} + \frac {1}{2} \, a^{3} c_{1} x^{2} + a^{3} b_{1} x \] Input:
integrate((c1*x+b1)*(c*x^2+2*b*x+a)^3,x, algorithm="giac")
Output:
1/8*c^3*c1*x^8 + 1/7*b1*c^3*x^7 + 6/7*b*c^2*c1*x^7 + b*b1*c^2*x^6 + 2*b^2* c*c1*x^6 + 1/2*a*c^2*c1*x^6 + 12/5*b^2*b1*c*x^5 + 3/5*a*b1*c^2*x^5 + 8/5*b ^3*c1*x^5 + 12/5*a*b*c*c1*x^5 + 2*b^3*b1*x^4 + 3*a*b*b1*c*x^4 + 3*a*b^2*c1 *x^4 + 3/4*a^2*c*c1*x^4 + 4*a*b^2*b1*x^3 + a^2*b1*c*x^3 + 2*a^2*b*c1*x^3 + 3*a^2*b*b1*x^2 + 1/2*a^3*c1*x^2 + a^3*b1*x
Time = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.98 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^3 \, dx=x^7\,\left (\frac {b_{1}\,c^3}{7}+\frac {6\,b\,c_{1}\,c^2}{7}\right )+x^3\,\left (2\,c_{1}\,a^2\,b+b_{1}\,c\,a^2+4\,b_{1}\,a\,b^2\right )+x^6\,\left (2\,c_{1}\,b^2\,c+b_{1}\,b\,c^2+\frac {a\,c_{1}\,c^2}{2}\right )+x^4\,\left (\frac {3\,c\,c_{1}\,a^2}{4}+3\,c_{1}\,a\,b^2+3\,b_{1}\,c\,a\,b+2\,b_{1}\,b^3\right )+x^5\,\left (\frac {8\,c_{1}\,b^3}{5}+\frac {12\,b_{1}\,b^2\,c}{5}+\frac {12\,a\,c_{1}\,b\,c}{5}+\frac {3\,a\,b_{1}\,c^2}{5}\right )+x^2\,\left (\frac {c_{1}\,a^3}{2}+3\,b\,b_{1}\,a^2\right )+\frac {c^3\,c_{1}\,x^8}{8}+a^3\,b_{1}\,x \] Input:
int((b1 + c1*x)*(a + 2*b*x + c*x^2)^3,x)
Output:
x^7*((b1*c^3)/7 + (6*b*c^2*c1)/7) + x^3*(4*a*b^2*b1 + 2*a^2*b*c1 + a^2*b1* c) + x^6*((a*c^2*c1)/2 + b*b1*c^2 + 2*b^2*c*c1) + x^4*(2*b^3*b1 + 3*a*b^2* c1 + (3*a^2*c*c1)/4 + 3*a*b*b1*c) + x^5*((8*b^3*c1)/5 + (3*a*b1*c^2)/5 + ( 12*b^2*b1*c)/5 + (12*a*b*c*c1)/5) + x^2*((a^3*c1)/2 + 3*a^2*b*b1) + (c^3*c 1*x^8)/8 + a^3*b1*x
Time = 0.15 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.13 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^3 \, dx=\frac {x \left (35 c^{3} \mathit {c1} \,x^{7}+240 b \,c^{2} \mathit {c1} \,x^{6}+40 \mathit {b1} \,c^{3} x^{6}+140 a \,c^{2} \mathit {c1} \,x^{5}+560 b^{2} c \mathit {c1} \,x^{5}+280 b \mathit {b1} \,c^{2} x^{5}+672 a b c \mathit {c1} \,x^{4}+168 a \mathit {b1} \,c^{2} x^{4}+448 b^{3} \mathit {c1} \,x^{4}+672 b^{2} \mathit {b1} c \,x^{4}+210 a^{2} c \mathit {c1} \,x^{3}+840 a \,b^{2} \mathit {c1} \,x^{3}+840 a b \mathit {b1} c \,x^{3}+560 b^{3} \mathit {b1} \,x^{3}+560 a^{2} b \mathit {c1} \,x^{2}+280 a^{2} \mathit {b1} c \,x^{2}+1120 a \,b^{2} \mathit {b1} \,x^{2}+140 a^{3} \mathit {c1} x +840 a^{2} b \mathit {b1} x +280 a^{3} \mathit {b1} \right )}{280} \] Input:
int((c1*x+b1)*(c*x^2+2*b*x+a)^3,x)
Output:
(x*(280*a**3*b1 + 140*a**3*c1*x + 840*a**2*b*b1*x + 560*a**2*b*c1*x**2 + 2 80*a**2*b1*c*x**2 + 210*a**2*c*c1*x**3 + 1120*a*b**2*b1*x**2 + 840*a*b**2* c1*x**3 + 840*a*b*b1*c*x**3 + 672*a*b*c*c1*x**4 + 168*a*b1*c**2*x**4 + 140 *a*c**2*c1*x**5 + 560*b**3*b1*x**3 + 448*b**3*c1*x**4 + 672*b**2*b1*c*x**4 + 560*b**2*c*c1*x**5 + 280*b*b1*c**2*x**5 + 240*b*c**2*c1*x**6 + 40*b1*c* *3*x**6 + 35*c**3*c1*x**7))/280