Integrand size = 19, antiderivative size = 127 \[ \int (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} b^3 d^2 x^4+\frac {1}{5} b^2 d (3 c d+2 b e) x^5+\frac {1}{6} b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^7+\frac {1}{8} c^2 e (2 c d+3 b e) x^8+\frac {1}{9} c^3 e^2 x^9 \]
1/4*b^3*d^2*x^4+1/5*b^2*d*(2*b*e+3*c*d)*x^5+1/6*b*(b^2*e^2+6*b*c*d*e+3*c^2 *d^2)*x^6+1/7*c*(3*b^2*e^2+6*b*c*d*e+c^2*d^2)*x^7+1/8*c^2*e*(3*b*e+2*c*d)* x^8+1/9*c^3*e^2*x^9
Time = 0.03 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} b^3 d^2 x^4+\frac {1}{5} b^2 d (3 c d+2 b e) x^5+\frac {1}{6} b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^7+\frac {1}{8} c^2 e (2 c d+3 b e) x^8+\frac {1}{9} c^3 e^2 x^9 \]
(b^3*d^2*x^4)/4 + (b^2*d*(3*c*d + 2*b*e)*x^5)/5 + (b*(3*c^2*d^2 + 6*b*c*d* e + b^2*e^2)*x^6)/6 + (c*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2)*x^7)/7 + (c^2*e *(2*c*d + 3*b*e)*x^8)/8 + (c^3*e^2*x^9)/9
Time = 0.30 (sec) , antiderivative size = 127, 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.
\(\displaystyle \int \left (b x+c x^2\right )^3 (d+e x)^2 \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (b^3 d^2 x^3+c x^6 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+b x^5 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+b^2 d x^4 (2 b e+3 c d)+c^2 e x^7 (3 b e+2 c d)+c^3 e^2 x^8\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} b^3 d^2 x^4+\frac {1}{7} c x^7 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac {1}{6} b x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac {1}{5} b^2 d x^5 (2 b e+3 c d)+\frac {1}{8} c^2 e x^8 (3 b e+2 c d)+\frac {1}{9} c^3 e^2 x^9\) |
(b^3*d^2*x^4)/4 + (b^2*d*(3*c*d + 2*b*e)*x^5)/5 + (b*(3*c^2*d^2 + 6*b*c*d* e + b^2*e^2)*x^6)/6 + (c*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2)*x^7)/7 + (c^2*e *(2*c*d + 3*b*e)*x^8)/8 + (c^3*e^2*x^9)/9
3.3.46.3.1 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 = 2.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.98
method | result | size |
norman | \(\frac {c^{3} e^{2} x^{9}}{9}+\left (\frac {3}{8} e^{2} b \,c^{2}+\frac {1}{4} d e \,c^{3}\right ) x^{8}+\left (\frac {3}{7} b^{2} e^{2} c +\frac {6}{7} d e b \,c^{2}+\frac {1}{7} c^{3} d^{2}\right ) x^{7}+\left (\frac {1}{6} e^{2} b^{3}+b^{2} d c e +\frac {1}{2} b \,c^{2} d^{2}\right ) x^{6}+\left (\frac {2}{5} b^{3} d e +\frac {3}{5} b^{2} c \,d^{2}\right ) x^{5}+\frac {d^{2} x^{4} b^{3}}{4}\) | \(125\) |
default | \(\frac {c^{3} e^{2} x^{9}}{9}+\frac {\left (3 e^{2} b \,c^{2}+2 d e \,c^{3}\right ) x^{8}}{8}+\frac {\left (3 b^{2} e^{2} c +6 d e b \,c^{2}+c^{3} d^{2}\right ) x^{7}}{7}+\frac {\left (e^{2} b^{3}+6 b^{2} d c e +3 b \,c^{2} d^{2}\right ) x^{6}}{6}+\frac {\left (2 b^{3} d e +3 b^{2} c \,d^{2}\right ) x^{5}}{5}+\frac {d^{2} x^{4} b^{3}}{4}\) | \(128\) |
gosper | \(\frac {x^{4} \left (280 c^{3} e^{2} x^{5}+945 x^{4} e^{2} b \,c^{2}+630 x^{4} d e \,c^{3}+1080 x^{3} b^{2} e^{2} c +2160 x^{3} d e b \,c^{2}+360 x^{3} c^{3} d^{2}+420 x^{2} e^{2} b^{3}+2520 x^{2} b^{2} d c e +1260 b \,c^{2} d^{2} x^{2}+1008 x \,b^{3} d e +1512 c \,d^{2} x \,b^{2}+630 b^{3} d^{2}\right )}{2520}\) | \(134\) |
risch | \(\frac {1}{9} c^{3} e^{2} x^{9}+\frac {3}{8} x^{8} e^{2} b \,c^{2}+\frac {1}{4} x^{8} d e \,c^{3}+\frac {3}{7} x^{7} b^{2} e^{2} c +\frac {6}{7} x^{7} d e b \,c^{2}+\frac {1}{7} x^{7} c^{3} d^{2}+\frac {1}{6} x^{6} e^{2} b^{3}+x^{6} b^{2} d c e +\frac {1}{2} x^{6} b \,c^{2} d^{2}+\frac {2}{5} x^{5} b^{3} d e +\frac {3}{5} b^{2} c \,d^{2} x^{5}+\frac {1}{4} d^{2} x^{4} b^{3}\) | \(135\) |
parallelrisch | \(\frac {1}{9} c^{3} e^{2} x^{9}+\frac {3}{8} x^{8} e^{2} b \,c^{2}+\frac {1}{4} x^{8} d e \,c^{3}+\frac {3}{7} x^{7} b^{2} e^{2} c +\frac {6}{7} x^{7} d e b \,c^{2}+\frac {1}{7} x^{7} c^{3} d^{2}+\frac {1}{6} x^{6} e^{2} b^{3}+x^{6} b^{2} d c e +\frac {1}{2} x^{6} b \,c^{2} d^{2}+\frac {2}{5} x^{5} b^{3} d e +\frac {3}{5} b^{2} c \,d^{2} x^{5}+\frac {1}{4} d^{2} x^{4} b^{3}\) | \(135\) |
1/9*c^3*e^2*x^9+(3/8*e^2*b*c^2+1/4*d*e*c^3)*x^8+(3/7*b^2*e^2*c+6/7*d*e*b*c ^2+1/7*c^3*d^2)*x^7+(1/6*e^2*b^3+b^2*d*c*e+1/2*b*c^2*d^2)*x^6+(2/5*b^3*d*e +3/5*b^2*c*d^2)*x^5+1/4*d^2*x^4*b^3
Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} e^{2} x^{9} + \frac {1}{4} \, b^{3} d^{2} x^{4} + \frac {1}{8} \, {\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, b c^{2} d^{2} + 6 \, b^{2} c d e + b^{3} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, b^{2} c d^{2} + 2 \, b^{3} d e\right )} x^{5} \]
1/9*c^3*e^2*x^9 + 1/4*b^3*d^2*x^4 + 1/8*(2*c^3*d*e + 3*b*c^2*e^2)*x^8 + 1/ 7*(c^3*d^2 + 6*b*c^2*d*e + 3*b^2*c*e^2)*x^7 + 1/6*(3*b*c^2*d^2 + 6*b^2*c*d *e + b^3*e^2)*x^6 + 1/5*(3*b^2*c*d^2 + 2*b^3*d*e)*x^5
Time = 0.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.09 \[ \int (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=\frac {b^{3} d^{2} x^{4}}{4} + \frac {c^{3} e^{2} x^{9}}{9} + x^{8} \cdot \left (\frac {3 b c^{2} e^{2}}{8} + \frac {c^{3} d e}{4}\right ) + x^{7} \cdot \left (\frac {3 b^{2} c e^{2}}{7} + \frac {6 b c^{2} d e}{7} + \frac {c^{3} d^{2}}{7}\right ) + x^{6} \left (\frac {b^{3} e^{2}}{6} + b^{2} c d e + \frac {b c^{2} d^{2}}{2}\right ) + x^{5} \cdot \left (\frac {2 b^{3} d e}{5} + \frac {3 b^{2} c d^{2}}{5}\right ) \]
b**3*d**2*x**4/4 + c**3*e**2*x**9/9 + x**8*(3*b*c**2*e**2/8 + c**3*d*e/4) + x**7*(3*b**2*c*e**2/7 + 6*b*c**2*d*e/7 + c**3*d**2/7) + x**6*(b**3*e**2/ 6 + b**2*c*d*e + b*c**2*d**2/2) + x**5*(2*b**3*d*e/5 + 3*b**2*c*d**2/5)
Time = 0.20 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} e^{2} x^{9} + \frac {1}{4} \, b^{3} d^{2} x^{4} + \frac {1}{8} \, {\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, b c^{2} d^{2} + 6 \, b^{2} c d e + b^{3} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, b^{2} c d^{2} + 2 \, b^{3} d e\right )} x^{5} \]
1/9*c^3*e^2*x^9 + 1/4*b^3*d^2*x^4 + 1/8*(2*c^3*d*e + 3*b*c^2*e^2)*x^8 + 1/ 7*(c^3*d^2 + 6*b*c^2*d*e + 3*b^2*c*e^2)*x^7 + 1/6*(3*b*c^2*d^2 + 6*b^2*c*d *e + b^3*e^2)*x^6 + 1/5*(3*b^2*c*d^2 + 2*b^3*d*e)*x^5
Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06 \[ \int (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} e^{2} x^{9} + \frac {1}{4} \, c^{3} d e x^{8} + \frac {3}{8} \, b c^{2} e^{2} x^{8} + \frac {1}{7} \, c^{3} d^{2} x^{7} + \frac {6}{7} \, b c^{2} d e x^{7} + \frac {3}{7} \, b^{2} c e^{2} x^{7} + \frac {1}{2} \, b c^{2} d^{2} x^{6} + b^{2} c d e x^{6} + \frac {1}{6} \, b^{3} e^{2} x^{6} + \frac {3}{5} \, b^{2} c d^{2} x^{5} + \frac {2}{5} \, b^{3} d e x^{5} + \frac {1}{4} \, b^{3} d^{2} x^{4} \]
1/9*c^3*e^2*x^9 + 1/4*c^3*d*e*x^8 + 3/8*b*c^2*e^2*x^8 + 1/7*c^3*d^2*x^7 + 6/7*b*c^2*d*e*x^7 + 3/7*b^2*c*e^2*x^7 + 1/2*b*c^2*d^2*x^6 + b^2*c*d*e*x^6 + 1/6*b^3*e^2*x^6 + 3/5*b^2*c*d^2*x^5 + 2/5*b^3*d*e*x^5 + 1/4*b^3*d^2*x^4
Time = 0.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.93 \[ \int (d+e x)^2 \left (b x+c x^2\right )^3 \, dx=x^6\,\left (\frac {b^3\,e^2}{6}+b^2\,c\,d\,e+\frac {b\,c^2\,d^2}{2}\right )+x^7\,\left (\frac {3\,b^2\,c\,e^2}{7}+\frac {6\,b\,c^2\,d\,e}{7}+\frac {c^3\,d^2}{7}\right )+\frac {b^3\,d^2\,x^4}{4}+\frac {c^3\,e^2\,x^9}{9}+\frac {b^2\,d\,x^5\,\left (2\,b\,e+3\,c\,d\right )}{5}+\frac {c^2\,e\,x^8\,\left (3\,b\,e+2\,c\,d\right )}{8} \]