Integrand size = 18, antiderivative size = 161 \[ \int (d+e x) \left (a+b x+c x^2\right )^3 \, dx=a^3 d x+\frac {1}{2} a^2 (3 b d+a e) x^2+a \left (b^2 d+a c d+a b e\right ) x^3+\frac {1}{4} \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) x^4+\frac {1}{5} \left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) x^5+\frac {1}{2} c \left (b c d+b^2 e+a c e\right ) x^6+\frac {1}{7} c^2 (c d+3 b e) x^7+\frac {1}{8} c^3 e x^8 \] Output:
a^3*d*x+1/2*a^2*(a*e+3*b*d)*x^2+a*(a*b*e+a*c*d+b^2*d)*x^3+1/4*(3*a^2*c*e+3 *a*b^2*e+6*a*b*c*d+b^3*d)*x^4+1/5*(6*a*b*c*e+3*a*c^2*d+b^3*e+3*b^2*c*d)*x^ 5+1/2*c*(a*c*e+b^2*e+b*c*d)*x^6+1/7*c^2*(3*b*e+c*d)*x^7+1/8*c^3*e*x^8
Time = 0.03 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00 \[ \int (d+e x) \left (a+b x+c x^2\right )^3 \, dx=a^3 d x+\frac {1}{2} a^2 (3 b d+a e) x^2+a \left (b^2 d+a c d+a b e\right ) x^3+\frac {1}{4} \left (b^3 d+6 a b c d+3 a b^2 e+3 a^2 c e\right ) x^4+\frac {1}{5} \left (3 b^2 c d+3 a c^2 d+b^3 e+6 a b c e\right ) x^5+\frac {1}{2} c \left (b c d+b^2 e+a c e\right ) x^6+\frac {1}{7} c^2 (c d+3 b e) x^7+\frac {1}{8} c^3 e x^8 \] Input:
Integrate[(d + e*x)*(a + b*x + c*x^2)^3,x]
Output:
a^3*d*x + (a^2*(3*b*d + a*e)*x^2)/2 + a*(b^2*d + a*c*d + a*b*e)*x^3 + ((b^ 3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c*e)*x^4)/4 + ((3*b^2*c*d + 3*a*c^2*d + b^3*e + 6*a*b*c*e)*x^5)/5 + (c*(b*c*d + b^2*e + a*c*e)*x^6)/2 + (c^2*(c* d + 3*b*e)*x^7)/7 + (c^3*e*x^8)/8
Time = 0.37 (sec) , antiderivative size = 161, 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.
| \(\displaystyle \int (d+e x) \left (a+b x+c x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (a^3 d+x^3 \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+a^2 x (a e+3 b d)+3 c x^5 \left (a c e+b^2 e+b c d\right )+3 a x^2 \left (a b e+a c d+b^2 d\right )+x^4 \left (6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right )+c^2 x^6 (3 b e+c d)+c^3 e x^7\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^3 d x+\frac {1}{4} x^4 \left (3 a^2 c e+3 a b^2 e+6 a b c d+b^3 d\right )+\frac {1}{2} a^2 x^2 (a e+3 b d)+\frac {1}{2} c x^6 \left (a c e+b^2 e+b c d\right )+a x^3 \left (a b e+a c d+b^2 d\right )+\frac {1}{5} x^5 \left (6 a b c e+3 a c^2 d+b^3 e+3 b^2 c d\right )+\frac {1}{7} c^2 x^7 (3 b e+c d)+\frac {1}{8} c^3 e x^8\) |
Input:
Int[(d + e*x)*(a + b*x + c*x^2)^3,x]
Output:
a^3*d*x + (a^2*(3*b*d + a*e)*x^2)/2 + a*(b^2*d + a*c*d + a*b*e)*x^3 + ((b^ 3*d + 6*a*b*c*d + 3*a*b^2*e + 3*a^2*c*e)*x^4)/4 + ((3*b^2*c*d + 3*a*c^2*d + b^3*e + 6*a*b*c*e)*x^5)/5 + (c*(b*c*d + b^2*e + a*c*e)*x^6)/2 + (c^2*(c* d + 3*b*e)*x^7)/7 + (c^3*e*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 = 0.77 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.02
| method | result | size |
| norman | \(\frac {c^{3} e \,x^{8}}{8}+\left (\frac {3}{7} b \,c^{2} e +\frac {1}{7} d \,c^{3}\right ) x^{7}+\left (\frac {1}{2} c^{2} a e +\frac {1}{2} c e \,b^{2}+\frac {1}{2} b \,c^{2} d \right ) x^{6}+\left (\frac {6}{5} a b c e +\frac {3}{5} a \,c^{2} d +\frac {1}{5} e \,b^{3}+\frac {3}{5} c d \,b^{2}\right ) x^{5}+\left (\frac {3}{4} a^{2} c e +\frac {3}{4} e a \,b^{2}+\frac {3}{2} a b c d +\frac {1}{4} b^{3} d \right ) x^{4}+\left (e \,a^{2} b +a^{2} c d +d a \,b^{2}\right ) x^{3}+\left (\frac {1}{2} a^{3} e +\frac {3}{2} a^{2} b d \right ) x^{2}+a^{3} d x\) | \(164\) |
| gosper | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {3}{7} x^{7} b \,c^{2} e +\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {1}{2} x^{6} c e \,b^{2}+\frac {1}{2} x^{6} b \,c^{2} d +\frac {6}{5} x^{5} a b c e +\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {1}{5} x^{5} e \,b^{3}+\frac {3}{5} b^{2} c d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+\frac {3}{4} x^{4} a \,b^{2} e +\frac {3}{2} a b c d \,x^{4}+\frac {1}{4} b^{3} d \,x^{4}+a^{2} b e \,x^{3}+a^{2} c d \,x^{3}+a \,b^{2} d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+\frac {3}{2} a^{2} b d \,x^{2}+a^{3} d x\) | \(188\) |
| risch | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {3}{7} x^{7} b \,c^{2} e +\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {1}{2} x^{6} c e \,b^{2}+\frac {1}{2} x^{6} b \,c^{2} d +\frac {6}{5} x^{5} a b c e +\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {1}{5} x^{5} e \,b^{3}+\frac {3}{5} b^{2} c d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+\frac {3}{4} x^{4} a \,b^{2} e +\frac {3}{2} a b c d \,x^{4}+\frac {1}{4} b^{3} d \,x^{4}+a^{2} b e \,x^{3}+a^{2} c d \,x^{3}+a \,b^{2} d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+\frac {3}{2} a^{2} b d \,x^{2}+a^{3} d x\) | \(188\) |
| parallelrisch | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {3}{7} x^{7} b \,c^{2} e +\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {1}{2} x^{6} c e \,b^{2}+\frac {1}{2} x^{6} b \,c^{2} d +\frac {6}{5} x^{5} a b c e +\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {1}{5} x^{5} e \,b^{3}+\frac {3}{5} b^{2} c d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+\frac {3}{4} x^{4} a \,b^{2} e +\frac {3}{2} a b c d \,x^{4}+\frac {1}{4} b^{3} d \,x^{4}+a^{2} b e \,x^{3}+a^{2} c d \,x^{3}+a \,b^{2} d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+\frac {3}{2} a^{2} b d \,x^{2}+a^{3} d x\) | \(188\) |
| orering | \(\frac {x \left (35 c^{3} e \,x^{7}+120 b \,c^{2} e \,x^{6}+40 c^{3} d \,x^{6}+140 a \,c^{2} e \,x^{5}+140 b^{2} c e \,x^{5}+140 b \,c^{2} d \,x^{5}+336 a b c e \,x^{4}+168 a \,c^{2} d \,x^{4}+56 e \,b^{3} x^{4}+168 b^{2} c d \,x^{4}+210 a^{2} c e \,x^{3}+210 a \,b^{2} e \,x^{3}+420 a b c d \,x^{3}+70 b^{3} d \,x^{3}+280 e \,a^{2} b \,x^{2}+280 a^{2} c d \,x^{2}+280 a \,b^{2} d \,x^{2}+140 a^{3} e x +420 a^{2} b d x +280 a^{3} d \right )}{280}\) | \(190\) |
| default | \(\frac {c^{3} e \,x^{8}}{8}+\frac {\left (3 b \,c^{2} e +d \,c^{3}\right ) x^{7}}{7}+\frac {\left (3 b \,c^{2} d +e \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (d \left (a \,c^{2}+2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+e \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{5}}{5}+\frac {\left (d \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+e \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{4}}{4}+\frac {\left (d \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+3 e \,a^{2} b \right ) x^{3}}{3}+\frac {\left (a^{3} e +3 a^{2} b d \right ) x^{2}}{2}+a^{3} d x\) | \(223\) |
Input:
int((e*x+d)*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/8*c^3*e*x^8+(3/7*b*c^2*e+1/7*d*c^3)*x^7+(1/2*c^2*a*e+1/2*c*e*b^2+1/2*b*c ^2*d)*x^6+(6/5*a*b*c*e+3/5*a*c^2*d+1/5*e*b^3+3/5*c*d*b^2)*x^5+(3/4*a^2*c*e +3/4*e*a*b^2+3/2*a*b*c*d+1/4*b^3*d)*x^4+(a^2*b*e+a^2*c*d+a*b^2*d)*x^3+(1/2 *a^3*e+3/2*a^2*b*d)*x^2+a^3*d*x
Time = 0.07 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.01 \[ \int (d+e x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} e x^{8} + \frac {1}{7} \, {\left (c^{3} d + 3 \, b c^{2} e\right )} x^{7} + \frac {1}{2} \, {\left (b c^{2} d + {\left (b^{2} c + a c^{2}\right )} e\right )} x^{6} + \frac {1}{5} \, {\left (3 \, {\left (b^{2} c + a c^{2}\right )} d + {\left (b^{3} + 6 \, a b c\right )} e\right )} x^{5} + a^{3} d x + \frac {1}{4} \, {\left ({\left (b^{3} + 6 \, a b c\right )} d + 3 \, {\left (a b^{2} + a^{2} c\right )} e\right )} x^{4} + {\left (a^{2} b e + {\left (a b^{2} + a^{2} c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d + a^{3} e\right )} x^{2} \] Input:
integrate((e*x+d)*(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
1/8*c^3*e*x^8 + 1/7*(c^3*d + 3*b*c^2*e)*x^7 + 1/2*(b*c^2*d + (b^2*c + a*c^ 2)*e)*x^6 + 1/5*(3*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e)*x^5 + a^3*d*x + 1/4*((b^3 + 6*a*b*c)*d + 3*(a*b^2 + a^2*c)*e)*x^4 + (a^2*b*e + (a*b^2 + a^ 2*c)*d)*x^3 + 1/2*(3*a^2*b*d + a^3*e)*x^2
Time = 0.05 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.18 \[ \int (d+e x) \left (a+b x+c x^2\right )^3 \, dx=a^{3} d x + \frac {c^{3} e x^{8}}{8} + x^{7} \cdot \left (\frac {3 b c^{2} e}{7} + \frac {c^{3} d}{7}\right ) + x^{6} \left (\frac {a c^{2} e}{2} + \frac {b^{2} c e}{2} + \frac {b c^{2} d}{2}\right ) + x^{5} \cdot \left (\frac {6 a b c e}{5} + \frac {3 a c^{2} d}{5} + \frac {b^{3} e}{5} + \frac {3 b^{2} c d}{5}\right ) + x^{4} \cdot \left (\frac {3 a^{2} c e}{4} + \frac {3 a b^{2} e}{4} + \frac {3 a b c d}{2} + \frac {b^{3} d}{4}\right ) + x^{3} \left (a^{2} b e + a^{2} c d + a b^{2} d\right ) + x^{2} \left (\frac {a^{3} e}{2} + \frac {3 a^{2} b d}{2}\right ) \] Input:
integrate((e*x+d)*(c*x**2+b*x+a)**3,x)
Output:
a**3*d*x + c**3*e*x**8/8 + x**7*(3*b*c**2*e/7 + c**3*d/7) + x**6*(a*c**2*e /2 + b**2*c*e/2 + b*c**2*d/2) + x**5*(6*a*b*c*e/5 + 3*a*c**2*d/5 + b**3*e/ 5 + 3*b**2*c*d/5) + x**4*(3*a**2*c*e/4 + 3*a*b**2*e/4 + 3*a*b*c*d/2 + b**3 *d/4) + x**3*(a**2*b*e + a**2*c*d + a*b**2*d) + x**2*(a**3*e/2 + 3*a**2*b* d/2)
Time = 0.03 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.01 \[ \int (d+e x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} e x^{8} + \frac {1}{7} \, {\left (c^{3} d + 3 \, b c^{2} e\right )} x^{7} + \frac {1}{2} \, {\left (b c^{2} d + {\left (b^{2} c + a c^{2}\right )} e\right )} x^{6} + \frac {1}{5} \, {\left (3 \, {\left (b^{2} c + a c^{2}\right )} d + {\left (b^{3} + 6 \, a b c\right )} e\right )} x^{5} + a^{3} d x + \frac {1}{4} \, {\left ({\left (b^{3} + 6 \, a b c\right )} d + 3 \, {\left (a b^{2} + a^{2} c\right )} e\right )} x^{4} + {\left (a^{2} b e + {\left (a b^{2} + a^{2} c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d + a^{3} e\right )} x^{2} \] Input:
integrate((e*x+d)*(c*x^2+b*x+a)^3,x, algorithm="maxima")
Output:
1/8*c^3*e*x^8 + 1/7*(c^3*d + 3*b*c^2*e)*x^7 + 1/2*(b*c^2*d + (b^2*c + a*c^ 2)*e)*x^6 + 1/5*(3*(b^2*c + a*c^2)*d + (b^3 + 6*a*b*c)*e)*x^5 + a^3*d*x + 1/4*((b^3 + 6*a*b*c)*d + 3*(a*b^2 + a^2*c)*e)*x^4 + (a^2*b*e + (a*b^2 + a^ 2*c)*d)*x^3 + 1/2*(3*a^2*b*d + a^3*e)*x^2
Time = 0.35 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.16 \[ \int (d+e x) \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} e x^{8} + \frac {1}{7} \, c^{3} d x^{7} + \frac {3}{7} \, b c^{2} e x^{7} + \frac {1}{2} \, b c^{2} d x^{6} + \frac {1}{2} \, b^{2} c e x^{6} + \frac {1}{2} \, a c^{2} e x^{6} + \frac {3}{5} \, b^{2} c d x^{5} + \frac {3}{5} \, a c^{2} d x^{5} + \frac {1}{5} \, b^{3} e x^{5} + \frac {6}{5} \, a b c e x^{5} + \frac {1}{4} \, b^{3} d x^{4} + \frac {3}{2} \, a b c d x^{4} + \frac {3}{4} \, a b^{2} e x^{4} + \frac {3}{4} \, a^{2} c e x^{4} + a b^{2} d x^{3} + a^{2} c d x^{3} + a^{2} b e x^{3} + \frac {3}{2} \, a^{2} b d x^{2} + \frac {1}{2} \, a^{3} e x^{2} + a^{3} d x \] Input:
integrate((e*x+d)*(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
1/8*c^3*e*x^8 + 1/7*c^3*d*x^7 + 3/7*b*c^2*e*x^7 + 1/2*b*c^2*d*x^6 + 1/2*b^ 2*c*e*x^6 + 1/2*a*c^2*e*x^6 + 3/5*b^2*c*d*x^5 + 3/5*a*c^2*d*x^5 + 1/5*b^3* e*x^5 + 6/5*a*b*c*e*x^5 + 1/4*b^3*d*x^4 + 3/2*a*b*c*d*x^4 + 3/4*a*b^2*e*x^ 4 + 3/4*a^2*c*e*x^4 + a*b^2*d*x^3 + a^2*c*d*x^3 + a^2*b*e*x^3 + 3/2*a^2*b* d*x^2 + 1/2*a^3*e*x^2 + a^3*d*x
Time = 0.04 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.01 \[ \int (d+e x) \left (a+b x+c x^2\right )^3 \, dx=x^2\,\left (\frac {e\,a^3}{2}+\frac {3\,b\,d\,a^2}{2}\right )+x^7\,\left (\frac {d\,c^3}{7}+\frac {3\,b\,e\,c^2}{7}\right )+x^3\,\left (e\,a^2\,b+c\,d\,a^2+d\,a\,b^2\right )+x^6\,\left (\frac {e\,b^2\,c}{2}+\frac {d\,b\,c^2}{2}+\frac {a\,e\,c^2}{2}\right )+x^4\,\left (\frac {3\,c\,e\,a^2}{4}+\frac {3\,e\,a\,b^2}{4}+\frac {3\,c\,d\,a\,b}{2}+\frac {d\,b^3}{4}\right )+x^5\,\left (\frac {e\,b^3}{5}+\frac {3\,d\,b^2\,c}{5}+\frac {6\,a\,e\,b\,c}{5}+\frac {3\,a\,d\,c^2}{5}\right )+\frac {c^3\,e\,x^8}{8}+a^3\,d\,x \] Input:
int((d + e*x)*(a + b*x + c*x^2)^3,x)
Output:
x^2*((a^3*e)/2 + (3*a^2*b*d)/2) + x^7*((c^3*d)/7 + (3*b*c^2*e)/7) + x^3*(a *b^2*d + a^2*b*e + a^2*c*d) + x^6*((a*c^2*e)/2 + (b*c^2*d)/2 + (b^2*c*e)/2 ) + x^4*((b^3*d)/4 + (3*a*b^2*e)/4 + (3*a^2*c*e)/4 + (3*a*b*c*d)/2) + x^5* ((b^3*e)/5 + (3*a*c^2*d)/5 + (3*b^2*c*d)/5 + (6*a*b*c*e)/5) + (c^3*e*x^8)/ 8 + a^3*d*x
Time = 0.25 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.17 \[ \int (d+e x) \left (a+b x+c x^2\right )^3 \, dx=\frac {x \left (35 c^{3} e \,x^{7}+120 b \,c^{2} e \,x^{6}+40 c^{3} d \,x^{6}+140 a \,c^{2} e \,x^{5}+140 b^{2} c e \,x^{5}+140 b \,c^{2} d \,x^{5}+336 a b c e \,x^{4}+168 a \,c^{2} d \,x^{4}+56 b^{3} e \,x^{4}+168 b^{2} c d \,x^{4}+210 a^{2} c e \,x^{3}+210 a \,b^{2} e \,x^{3}+420 a b c d \,x^{3}+70 b^{3} d \,x^{3}+280 a^{2} b e \,x^{2}+280 a^{2} c d \,x^{2}+280 a \,b^{2} d \,x^{2}+140 a^{3} e x +420 a^{2} b d x +280 a^{3} d \right )}{280} \] Input:
int((e*x+d)*(c*x^2+b*x+a)^3,x)
Output:
(x*(280*a**3*d + 140*a**3*e*x + 420*a**2*b*d*x + 280*a**2*b*e*x**2 + 280*a **2*c*d*x**2 + 210*a**2*c*e*x**3 + 280*a*b**2*d*x**2 + 210*a*b**2*e*x**3 + 420*a*b*c*d*x**3 + 336*a*b*c*e*x**4 + 168*a*c**2*d*x**4 + 140*a*c**2*e*x* *5 + 70*b**3*d*x**3 + 56*b**3*e*x**4 + 168*b**2*c*d*x**4 + 140*b**2*c*e*x* *5 + 140*b*c**2*d*x**5 + 120*b*c**2*e*x**6 + 40*c**3*d*x**6 + 35*c**3*e*x* *7))/280