Integrand size = 20, antiderivative size = 156 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}{4 e^5}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^5}{5 e^5}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^6}{6 e^5}-\frac {2 c (2 c d-b e) (d+e x)^7}{7 e^5}+\frac {c^2 (d+e x)^8}{8 e^5} \]
1/4*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^4/e^5-2/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^ 2)*(e*x+d)^5/e^5+1/6*(6*c^2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))*(e*x+d)^6/e^5- 2/7*c*(-b*e+2*c*d)*(e*x+d)^7/e^5+1/8*c^2*(e*x+d)^8/e^5
Time = 0.04 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.43 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=a^2 d^3 x+\frac {1}{2} a d^2 (2 b d+3 a e) x^2+\frac {1}{3} d \left (b^2 d^2+6 a b d e+a \left (2 c d^2+3 a e^2\right )\right ) x^3+\frac {1}{4} \left (2 b c d^3+3 b^2 d^2 e+6 a c d^2 e+6 a b d e^2+a^2 e^3\right ) x^4+\frac {1}{5} \left (c^2 d^3+6 c d e (b d+a e)+b e^2 (3 b d+2 a e)\right ) x^5+\frac {1}{6} e \left (3 c^2 d^2+b^2 e^2+2 c e (3 b d+a e)\right ) x^6+\frac {1}{7} c e^2 (3 c d+2 b e) x^7+\frac {1}{8} c^2 e^3 x^8 \]
a^2*d^3*x + (a*d^2*(2*b*d + 3*a*e)*x^2)/2 + (d*(b^2*d^2 + 6*a*b*d*e + a*(2 *c*d^2 + 3*a*e^2))*x^3)/3 + ((2*b*c*d^3 + 3*b^2*d^2*e + 6*a*c*d^2*e + 6*a* b*d*e^2 + a^2*e^3)*x^4)/4 + ((c^2*d^3 + 6*c*d*e*(b*d + a*e) + b*e^2*(3*b*d + 2*a*e))*x^5)/5 + (e*(3*c^2*d^2 + b^2*e^2 + 2*c*e*(3*b*d + a*e))*x^6)/6 + (c*e^2*(3*c*d + 2*b*e)*x^7)/7 + (c^2*e^3*x^8)/8
Time = 0.39 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac {2 (d+e x)^4 (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^4}+\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}{e^4}-\frac {2 c (d+e x)^6 (2 c d-b e)}{e^4}+\frac {c^2 (d+e x)^7}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^6 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^5}-\frac {2 (d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^5}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}{4 e^5}-\frac {2 c (d+e x)^7 (2 c d-b e)}{7 e^5}+\frac {c^2 (d+e x)^8}{8 e^5}\) |
((c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^4)/(4*e^5) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5)/(5*e^5) + ((6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b *d - a*e))*(d + e*x)^6)/(6*e^5) - (2*c*(2*c*d - b*e)*(d + e*x)^7)/(7*e^5) + (c^2*(d + e*x)^8)/(8*e^5)
3.22.19.3.1 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 = 2.99 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.40
| method | result | size |
| default | \(\frac {c^{2} e^{3} x^{8}}{8}+\frac {\left (2 e^{3} b c +3 d \,e^{2} c^{2}\right ) x^{7}}{7}+\frac {\left (3 d^{2} e \,c^{2}+6 b c d \,e^{2}+e^{3} \left (2 a c +b^{2}\right )\right ) x^{6}}{6}+\frac {\left (c^{2} d^{3}+6 b c e \,d^{2}+3 d \,e^{2} \left (2 a c +b^{2}\right )+2 a b \,e^{3}\right ) x^{5}}{5}+\frac {\left (2 b c \,d^{3}+3 d^{2} e \left (2 a c +b^{2}\right )+6 a b d \,e^{2}+a^{2} e^{3}\right ) x^{4}}{4}+\frac {\left (d^{3} \left (2 a c +b^{2}\right )+6 a b \,d^{2} e +3 a^{2} d \,e^{2}\right ) x^{3}}{3}+\frac {\left (3 a^{2} d^{2} e +2 a b \,d^{3}\right ) x^{2}}{2}+x \,a^{2} d^{3}\) | \(219\) |
| norman | \(\frac {c^{2} e^{3} x^{8}}{8}+\left (\frac {2}{7} e^{3} b c +\frac {3}{7} d \,e^{2} c^{2}\right ) x^{7}+\left (\frac {1}{3} a c \,e^{3}+\frac {1}{6} b^{2} e^{3}+b c d \,e^{2}+\frac {1}{2} d^{2} e \,c^{2}\right ) x^{6}+\left (\frac {2}{5} a b \,e^{3}+\frac {6}{5} d \,e^{2} a c +\frac {3}{5} b^{2} d \,e^{2}+\frac {6}{5} b c e \,d^{2}+\frac {1}{5} c^{2} d^{3}\right ) x^{5}+\left (\frac {1}{4} a^{2} e^{3}+\frac {3}{2} a b d \,e^{2}+\frac {3}{2} d^{2} e a c +\frac {3}{4} b^{2} d^{2} e +\frac {1}{2} b c \,d^{3}\right ) x^{4}+\left (a^{2} d \,e^{2}+2 a b \,d^{2} e +\frac {2}{3} a c \,d^{3}+\frac {1}{3} b^{2} d^{3}\right ) x^{3}+\left (\frac {3}{2} a^{2} d^{2} e +a b \,d^{3}\right ) x^{2}+x \,a^{2} d^{3}\) | \(224\) |
| gosper | \(\frac {1}{8} c^{2} e^{3} x^{8}+\frac {2}{7} x^{7} e^{3} b c +\frac {3}{7} d \,e^{2} c^{2} x^{7}+\frac {1}{3} x^{6} a c \,e^{3}+\frac {1}{6} b^{2} e^{3} x^{6}+x^{6} b c d \,e^{2}+\frac {1}{2} x^{6} d^{2} e \,c^{2}+\frac {2}{5} x^{5} a b \,e^{3}+\frac {6}{5} x^{5} d \,e^{2} a c +\frac {3}{5} x^{5} b^{2} d \,e^{2}+\frac {6}{5} x^{5} b c e \,d^{2}+\frac {1}{5} x^{5} c^{2} d^{3}+\frac {1}{4} x^{4} a^{2} e^{3}+\frac {3}{2} x^{4} a b d \,e^{2}+\frac {3}{2} x^{4} d^{2} e a c +\frac {3}{4} x^{4} b^{2} d^{2} e +\frac {1}{2} x^{4} b c \,d^{3}+x^{3} a^{2} d \,e^{2}+2 x^{3} a b \,d^{2} e +\frac {2}{3} a c \,d^{3} x^{3}+\frac {1}{3} b^{2} d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{2} x^{2}+x^{2} a b \,d^{3}+x \,a^{2} d^{3}\) | \(260\) |
| risch | \(\frac {1}{8} c^{2} e^{3} x^{8}+\frac {2}{7} x^{7} e^{3} b c +\frac {3}{7} d \,e^{2} c^{2} x^{7}+\frac {1}{3} x^{6} a c \,e^{3}+\frac {1}{6} b^{2} e^{3} x^{6}+x^{6} b c d \,e^{2}+\frac {1}{2} x^{6} d^{2} e \,c^{2}+\frac {2}{5} x^{5} a b \,e^{3}+\frac {6}{5} x^{5} d \,e^{2} a c +\frac {3}{5} x^{5} b^{2} d \,e^{2}+\frac {6}{5} x^{5} b c e \,d^{2}+\frac {1}{5} x^{5} c^{2} d^{3}+\frac {1}{4} x^{4} a^{2} e^{3}+\frac {3}{2} x^{4} a b d \,e^{2}+\frac {3}{2} x^{4} d^{2} e a c +\frac {3}{4} x^{4} b^{2} d^{2} e +\frac {1}{2} x^{4} b c \,d^{3}+x^{3} a^{2} d \,e^{2}+2 x^{3} a b \,d^{2} e +\frac {2}{3} a c \,d^{3} x^{3}+\frac {1}{3} b^{2} d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{2} x^{2}+x^{2} a b \,d^{3}+x \,a^{2} d^{3}\) | \(260\) |
| parallelrisch | \(\frac {1}{8} c^{2} e^{3} x^{8}+\frac {2}{7} x^{7} e^{3} b c +\frac {3}{7} d \,e^{2} c^{2} x^{7}+\frac {1}{3} x^{6} a c \,e^{3}+\frac {1}{6} b^{2} e^{3} x^{6}+x^{6} b c d \,e^{2}+\frac {1}{2} x^{6} d^{2} e \,c^{2}+\frac {2}{5} x^{5} a b \,e^{3}+\frac {6}{5} x^{5} d \,e^{2} a c +\frac {3}{5} x^{5} b^{2} d \,e^{2}+\frac {6}{5} x^{5} b c e \,d^{2}+\frac {1}{5} x^{5} c^{2} d^{3}+\frac {1}{4} x^{4} a^{2} e^{3}+\frac {3}{2} x^{4} a b d \,e^{2}+\frac {3}{2} x^{4} d^{2} e a c +\frac {3}{4} x^{4} b^{2} d^{2} e +\frac {1}{2} x^{4} b c \,d^{3}+x^{3} a^{2} d \,e^{2}+2 x^{3} a b \,d^{2} e +\frac {2}{3} a c \,d^{3} x^{3}+\frac {1}{3} b^{2} d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{2} x^{2}+x^{2} a b \,d^{3}+x \,a^{2} d^{3}\) | \(260\) |
1/8*c^2*e^3*x^8+1/7*(2*b*c*e^3+3*c^2*d*e^2)*x^7+1/6*(3*d^2*e*c^2+6*b*c*d*e ^2+e^3*(2*a*c+b^2))*x^6+1/5*(c^2*d^3+6*b*c*e*d^2+3*d*e^2*(2*a*c+b^2)+2*a*b *e^3)*x^5+1/4*(2*b*c*d^3+3*d^2*e*(2*a*c+b^2)+6*a*b*d*e^2+a^2*e^3)*x^4+1/3* (d^3*(2*a*c+b^2)+6*a*b*d^2*e+3*a^2*d*e^2)*x^3+1/2*(3*a^2*d^2*e+2*a*b*d^3)* x^2+x*a^2*d^3
Time = 0.30 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.40 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{8} \, c^{2} e^{3} x^{8} + \frac {1}{7} \, {\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{6} + a^{2} d^{3} x + \frac {1}{5} \, {\left (c^{2} d^{3} + 6 \, b c d^{2} e + 2 \, a b e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d^{3} + 6 \, a b d e^{2} + a^{2} e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a b d^{2} e + 3 \, a^{2} d e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{2} \]
1/8*c^2*e^3*x^8 + 1/7*(3*c^2*d*e^2 + 2*b*c*e^3)*x^7 + 1/6*(3*c^2*d^2*e + 6 *b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x^6 + a^2*d^3*x + 1/5*(c^2*d^3 + 6*b*c*d^2 *e + 2*a*b*e^3 + 3*(b^2 + 2*a*c)*d*e^2)*x^5 + 1/4*(2*b*c*d^3 + 6*a*b*d*e^2 + a^2*e^3 + 3*(b^2 + 2*a*c)*d^2*e)*x^4 + 1/3*(6*a*b*d^2*e + 3*a^2*d*e^2 + (b^2 + 2*a*c)*d^3)*x^3 + 1/2*(2*a*b*d^3 + 3*a^2*d^2*e)*x^2
Time = 0.03 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.67 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=a^{2} d^{3} x + \frac {c^{2} e^{3} x^{8}}{8} + x^{7} \cdot \left (\frac {2 b c e^{3}}{7} + \frac {3 c^{2} d e^{2}}{7}\right ) + x^{6} \left (\frac {a c e^{3}}{3} + \frac {b^{2} e^{3}}{6} + b c d e^{2} + \frac {c^{2} d^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {2 a b e^{3}}{5} + \frac {6 a c d e^{2}}{5} + \frac {3 b^{2} d e^{2}}{5} + \frac {6 b c d^{2} e}{5} + \frac {c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{2} e^{3}}{4} + \frac {3 a b d e^{2}}{2} + \frac {3 a c d^{2} e}{2} + \frac {3 b^{2} d^{2} e}{4} + \frac {b c d^{3}}{2}\right ) + x^{3} \left (a^{2} d e^{2} + 2 a b d^{2} e + \frac {2 a c d^{3}}{3} + \frac {b^{2} d^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 a^{2} d^{2} e}{2} + a b d^{3}\right ) \]
a**2*d**3*x + c**2*e**3*x**8/8 + x**7*(2*b*c*e**3/7 + 3*c**2*d*e**2/7) + x **6*(a*c*e**3/3 + b**2*e**3/6 + b*c*d*e**2 + c**2*d**2*e/2) + x**5*(2*a*b* e**3/5 + 6*a*c*d*e**2/5 + 3*b**2*d*e**2/5 + 6*b*c*d**2*e/5 + c**2*d**3/5) + x**4*(a**2*e**3/4 + 3*a*b*d*e**2/2 + 3*a*c*d**2*e/2 + 3*b**2*d**2*e/4 + b*c*d**3/2) + x**3*(a**2*d*e**2 + 2*a*b*d**2*e + 2*a*c*d**3/3 + b**2*d**3/ 3) + x**2*(3*a**2*d**2*e/2 + a*b*d**3)
Time = 0.20 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.40 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{8} \, c^{2} e^{3} x^{8} + \frac {1}{7} \, {\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{6} + a^{2} d^{3} x + \frac {1}{5} \, {\left (c^{2} d^{3} + 6 \, b c d^{2} e + 2 \, a b e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d^{3} + 6 \, a b d e^{2} + a^{2} e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a b d^{2} e + 3 \, a^{2} d e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{2} \]
1/8*c^2*e^3*x^8 + 1/7*(3*c^2*d*e^2 + 2*b*c*e^3)*x^7 + 1/6*(3*c^2*d^2*e + 6 *b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x^6 + a^2*d^3*x + 1/5*(c^2*d^3 + 6*b*c*d^2 *e + 2*a*b*e^3 + 3*(b^2 + 2*a*c)*d*e^2)*x^5 + 1/4*(2*b*c*d^3 + 6*a*b*d*e^2 + a^2*e^3 + 3*(b^2 + 2*a*c)*d^2*e)*x^4 + 1/3*(6*a*b*d^2*e + 3*a^2*d*e^2 + (b^2 + 2*a*c)*d^3)*x^3 + 1/2*(2*a*b*d^3 + 3*a^2*d^2*e)*x^2
Time = 0.29 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.66 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{8} \, c^{2} e^{3} x^{8} + \frac {3}{7} \, c^{2} d e^{2} x^{7} + \frac {2}{7} \, b c e^{3} x^{7} + \frac {1}{2} \, c^{2} d^{2} e x^{6} + b c d e^{2} x^{6} + \frac {1}{6} \, b^{2} e^{3} x^{6} + \frac {1}{3} \, a c e^{3} x^{6} + \frac {1}{5} \, c^{2} d^{3} x^{5} + \frac {6}{5} \, b c d^{2} e x^{5} + \frac {3}{5} \, b^{2} d e^{2} x^{5} + \frac {6}{5} \, a c d e^{2} x^{5} + \frac {2}{5} \, a b e^{3} x^{5} + \frac {1}{2} \, b c d^{3} x^{4} + \frac {3}{4} \, b^{2} d^{2} e x^{4} + \frac {3}{2} \, a c d^{2} e x^{4} + \frac {3}{2} \, a b d e^{2} x^{4} + \frac {1}{4} \, a^{2} e^{3} x^{4} + \frac {1}{3} \, b^{2} d^{3} x^{3} + \frac {2}{3} \, a c d^{3} x^{3} + 2 \, a b d^{2} e x^{3} + a^{2} d e^{2} x^{3} + a b d^{3} x^{2} + \frac {3}{2} \, a^{2} d^{2} e x^{2} + a^{2} d^{3} x \]
1/8*c^2*e^3*x^8 + 3/7*c^2*d*e^2*x^7 + 2/7*b*c*e^3*x^7 + 1/2*c^2*d^2*e*x^6 + b*c*d*e^2*x^6 + 1/6*b^2*e^3*x^6 + 1/3*a*c*e^3*x^6 + 1/5*c^2*d^3*x^5 + 6/ 5*b*c*d^2*e*x^5 + 3/5*b^2*d*e^2*x^5 + 6/5*a*c*d*e^2*x^5 + 2/5*a*b*e^3*x^5 + 1/2*b*c*d^3*x^4 + 3/4*b^2*d^2*e*x^4 + 3/2*a*c*d^2*e*x^4 + 3/2*a*b*d*e^2* x^4 + 1/4*a^2*e^3*x^4 + 1/3*b^2*d^3*x^3 + 2/3*a*c*d^3*x^3 + 2*a*b*d^2*e*x^ 3 + a^2*d*e^2*x^3 + a*b*d^3*x^2 + 3/2*a^2*d^2*e*x^2 + a^2*d^3*x
Time = 0.08 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.40 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx=x^4\,\left (\frac {a^2\,e^3}{4}+\frac {3\,a\,b\,d\,e^2}{2}+\frac {3\,c\,a\,d^2\,e}{2}+\frac {3\,b^2\,d^2\,e}{4}+\frac {c\,b\,d^3}{2}\right )+x^5\,\left (\frac {3\,b^2\,d\,e^2}{5}+\frac {6\,b\,c\,d^2\,e}{5}+\frac {2\,a\,b\,e^3}{5}+\frac {c^2\,d^3}{5}+\frac {6\,a\,c\,d\,e^2}{5}\right )+x^3\,\left (a^2\,d\,e^2+2\,a\,b\,d^2\,e+\frac {2\,c\,a\,d^3}{3}+\frac {b^2\,d^3}{3}\right )+x^6\,\left (\frac {b^2\,e^3}{6}+b\,c\,d\,e^2+\frac {c^2\,d^2\,e}{2}+\frac {a\,c\,e^3}{3}\right )+a^2\,d^3\,x+\frac {c^2\,e^3\,x^8}{8}+\frac {a\,d^2\,x^2\,\left (3\,a\,e+2\,b\,d\right )}{2}+\frac {c\,e^2\,x^7\,\left (2\,b\,e+3\,c\,d\right )}{7} \]
x^4*((a^2*e^3)/4 + (3*b^2*d^2*e)/4 + (b*c*d^3)/2 + (3*a*b*d*e^2)/2 + (3*a* c*d^2*e)/2) + x^5*((c^2*d^3)/5 + (3*b^2*d*e^2)/5 + (2*a*b*e^3)/5 + (6*a*c* d*e^2)/5 + (6*b*c*d^2*e)/5) + x^3*((b^2*d^3)/3 + a^2*d*e^2 + (2*a*c*d^3)/3 + 2*a*b*d^2*e) + x^6*((b^2*e^3)/6 + (c^2*d^2*e)/2 + (a*c*e^3)/3 + b*c*d*e ^2) + a^2*d^3*x + (c^2*e^3*x^8)/8 + (a*d^2*x^2*(3*a*e + 2*b*d))/2 + (c*e^2 *x^7*(2*b*e + 3*c*d))/7