Integrand size = 19, antiderivative size = 137 \[ \int (d+e x)^4 \left (b x+c x^2\right )^2 \, dx=\frac {d^2 (c d-b e)^2 (d+e x)^5}{5 e^5}-\frac {d (c d-b e) (2 c d-b e) (d+e x)^6}{3 e^5}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^7}{7 e^5}-\frac {c (2 c d-b e) (d+e x)^8}{4 e^5}+\frac {c^2 (d+e x)^9}{9 e^5} \]
1/5*d^2*(-b*e+c*d)^2*(e*x+d)^5/e^5-1/3*d*(-b*e+c*d)*(-b*e+2*c*d)*(e*x+d)^6 /e^5+1/7*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)*(e*x+d)^7/e^5-1/4*c*(-b*e+2*c*d)*(e *x+d)^8/e^5+1/9*c^2*(e*x+d)^9/e^5
Time = 0.03 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.16 \[ \int (d+e x)^4 \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} b^2 d^4 x^3+\frac {1}{2} b d^3 (c d+2 b e) x^4+\frac {1}{5} d^2 \left (c^2 d^2+8 b c d e+6 b^2 e^2\right ) x^5+\frac {2}{3} d e \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} e^2 \left (6 c^2 d^2+8 b c d e+b^2 e^2\right ) x^7+\frac {1}{4} c e^3 (2 c d+b e) x^8+\frac {1}{9} c^2 e^4 x^9 \]
(b^2*d^4*x^3)/3 + (b*d^3*(c*d + 2*b*e)*x^4)/2 + (d^2*(c^2*d^2 + 8*b*c*d*e + 6*b^2*e^2)*x^5)/5 + (2*d*e*(c^2*d^2 + 3*b*c*d*e + b^2*e^2)*x^6)/3 + (e^2 *(6*c^2*d^2 + 8*b*c*d*e + b^2*e^2)*x^7)/7 + (c*e^3*(2*c*d + b*e)*x^8)/4 + (c^2*e^4*x^9)/9
Time = 0.34 (sec) , antiderivative size = 137, 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.
\(\displaystyle \int \left (b x+c x^2\right )^2 (d+e x)^4 \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (\frac {(d+e x)^6 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{e^4}+\frac {d^2 (d+e x)^4 (c d-b e)^2}{e^4}-\frac {2 c (d+e x)^7 (2 c d-b e)}{e^4}+\frac {2 d (d+e x)^5 (c d-b e) (b e-2 c d)}{e^4}+\frac {c^2 (d+e x)^8}{e^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(d+e x)^7 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac {d^2 (d+e x)^5 (c d-b e)^2}{5 e^5}-\frac {c (d+e x)^8 (2 c d-b e)}{4 e^5}-\frac {d (d+e x)^6 (c d-b e) (2 c d-b e)}{3 e^5}+\frac {c^2 (d+e x)^9}{9 e^5}\) |
(d^2*(c*d - b*e)^2*(d + e*x)^5)/(5*e^5) - (d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^6)/(3*e^5) + ((6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^7)/(7*e^5 ) - (c*(2*c*d - b*e)*(d + e*x)^8)/(4*e^5) + (c^2*(d + e*x)^9)/(9*e^5)
3.3.31.3.1 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 = 1.86 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.18
method | result | size |
norman | \(\frac {e^{4} c^{2} x^{9}}{9}+\left (\frac {1}{4} e^{4} b c +\frac {1}{2} d \,e^{3} c^{2}\right ) x^{8}+\left (\frac {1}{7} e^{4} b^{2}+\frac {8}{7} d \,e^{3} b c +\frac {6}{7} d^{2} e^{2} c^{2}\right ) x^{7}+\left (\frac {2}{3} d \,e^{3} b^{2}+2 b c \,d^{2} e^{2}+\frac {2}{3} d^{3} e \,c^{2}\right ) x^{6}+\left (\frac {6}{5} b^{2} d^{2} e^{2}+\frac {8}{5} d^{3} e b c +\frac {1}{5} c^{2} d^{4}\right ) x^{5}+\left (d^{3} e \,b^{2}+\frac {1}{2} d^{4} b c \right ) x^{4}+\frac {d^{4} b^{2} x^{3}}{3}\) | \(162\) |
default | \(\frac {e^{4} c^{2} x^{9}}{9}+\frac {\left (2 e^{4} b c +4 d \,e^{3} c^{2}\right ) x^{8}}{8}+\frac {\left (e^{4} b^{2}+8 d \,e^{3} b c +6 d^{2} e^{2} c^{2}\right ) x^{7}}{7}+\frac {\left (4 d \,e^{3} b^{2}+12 b c \,d^{2} e^{2}+4 d^{3} e \,c^{2}\right ) x^{6}}{6}+\frac {\left (6 b^{2} d^{2} e^{2}+8 d^{3} e b c +c^{2} d^{4}\right ) x^{5}}{5}+\frac {\left (4 d^{3} e \,b^{2}+2 d^{4} b c \right ) x^{4}}{4}+\frac {d^{4} b^{2} x^{3}}{3}\) | \(166\) |
gosper | \(\frac {x^{3} \left (140 e^{4} c^{2} x^{6}+315 x^{5} e^{4} b c +630 x^{5} d \,e^{3} c^{2}+180 x^{4} e^{4} b^{2}+1440 x^{4} d \,e^{3} b c +1080 x^{4} d^{2} e^{2} c^{2}+840 x^{3} d \,e^{3} b^{2}+2520 x^{3} b c \,d^{2} e^{2}+840 x^{3} d^{3} e \,c^{2}+1512 x^{2} b^{2} d^{2} e^{2}+2016 x^{2} d^{3} e b c +252 x^{2} c^{2} d^{4}+1260 x \,d^{3} e \,b^{2}+630 x \,d^{4} b c +420 d^{4} b^{2}\right )}{1260}\) | \(175\) |
risch | \(\frac {1}{9} e^{4} c^{2} x^{9}+\frac {1}{4} x^{8} e^{4} b c +\frac {1}{2} x^{8} d \,e^{3} c^{2}+\frac {1}{7} x^{7} e^{4} b^{2}+\frac {8}{7} x^{7} d \,e^{3} b c +\frac {6}{7} x^{7} d^{2} e^{2} c^{2}+\frac {2}{3} x^{6} d \,e^{3} b^{2}+2 x^{6} b c \,d^{2} e^{2}+\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {6}{5} x^{5} b^{2} d^{2} e^{2}+\frac {8}{5} x^{5} d^{3} e b c +\frac {1}{5} c^{2} d^{4} x^{5}+x^{4} d^{3} e \,b^{2}+\frac {1}{2} x^{4} d^{4} b c +\frac {1}{3} d^{4} b^{2} x^{3}\) | \(176\) |
parallelrisch | \(\frac {1}{9} e^{4} c^{2} x^{9}+\frac {1}{4} x^{8} e^{4} b c +\frac {1}{2} x^{8} d \,e^{3} c^{2}+\frac {1}{7} x^{7} e^{4} b^{2}+\frac {8}{7} x^{7} d \,e^{3} b c +\frac {6}{7} x^{7} d^{2} e^{2} c^{2}+\frac {2}{3} x^{6} d \,e^{3} b^{2}+2 x^{6} b c \,d^{2} e^{2}+\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {6}{5} x^{5} b^{2} d^{2} e^{2}+\frac {8}{5} x^{5} d^{3} e b c +\frac {1}{5} c^{2} d^{4} x^{5}+x^{4} d^{3} e \,b^{2}+\frac {1}{2} x^{4} d^{4} b c +\frac {1}{3} d^{4} b^{2} x^{3}\) | \(176\) |
1/9*e^4*c^2*x^9+(1/4*e^4*b*c+1/2*d*e^3*c^2)*x^8+(1/7*e^4*b^2+8/7*d*e^3*b*c +6/7*d^2*e^2*c^2)*x^7+(2/3*d*e^3*b^2+2*b*c*d^2*e^2+2/3*d^3*e*c^2)*x^6+(6/5 *b^2*d^2*e^2+8/5*d^3*e*b*c+1/5*c^2*d^4)*x^5+(d^3*e*b^2+1/2*d^4*b*c)*x^4+1/ 3*d^4*b^2*x^3
Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.18 \[ \int (d+e x)^4 \left (b x+c x^2\right )^2 \, dx=\frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{3} \, b^{2} d^{4} x^{3} + \frac {1}{4} \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, c^{2} d^{2} e^{2} + 8 \, b c d e^{3} + b^{2} e^{4}\right )} x^{7} + \frac {2}{3} \, {\left (c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{4} + 8 \, b c d^{3} e + 6 \, b^{2} d^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{4} + 2 \, b^{2} d^{3} e\right )} x^{4} \]
1/9*c^2*e^4*x^9 + 1/3*b^2*d^4*x^3 + 1/4*(2*c^2*d*e^3 + b*c*e^4)*x^8 + 1/7* (6*c^2*d^2*e^2 + 8*b*c*d*e^3 + b^2*e^4)*x^7 + 2/3*(c^2*d^3*e + 3*b*c*d^2*e ^2 + b^2*d*e^3)*x^6 + 1/5*(c^2*d^4 + 8*b*c*d^3*e + 6*b^2*d^2*e^2)*x^5 + 1/ 2*(b*c*d^4 + 2*b^2*d^3*e)*x^4
Time = 0.04 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.30 \[ \int (d+e x)^4 \left (b x+c x^2\right )^2 \, dx=\frac {b^{2} d^{4} x^{3}}{3} + \frac {c^{2} e^{4} x^{9}}{9} + x^{8} \left (\frac {b c e^{4}}{4} + \frac {c^{2} d e^{3}}{2}\right ) + x^{7} \left (\frac {b^{2} e^{4}}{7} + \frac {8 b c d e^{3}}{7} + \frac {6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \cdot \left (\frac {2 b^{2} d e^{3}}{3} + 2 b c d^{2} e^{2} + \frac {2 c^{2} d^{3} e}{3}\right ) + x^{5} \cdot \left (\frac {6 b^{2} d^{2} e^{2}}{5} + \frac {8 b c d^{3} e}{5} + \frac {c^{2} d^{4}}{5}\right ) + x^{4} \left (b^{2} d^{3} e + \frac {b c d^{4}}{2}\right ) \]
b**2*d**4*x**3/3 + c**2*e**4*x**9/9 + x**8*(b*c*e**4/4 + c**2*d*e**3/2) + x**7*(b**2*e**4/7 + 8*b*c*d*e**3/7 + 6*c**2*d**2*e**2/7) + x**6*(2*b**2*d* e**3/3 + 2*b*c*d**2*e**2 + 2*c**2*d**3*e/3) + x**5*(6*b**2*d**2*e**2/5 + 8 *b*c*d**3*e/5 + c**2*d**4/5) + x**4*(b**2*d**3*e + b*c*d**4/2)
Time = 0.19 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.18 \[ \int (d+e x)^4 \left (b x+c x^2\right )^2 \, dx=\frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{3} \, b^{2} d^{4} x^{3} + \frac {1}{4} \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, c^{2} d^{2} e^{2} + 8 \, b c d e^{3} + b^{2} e^{4}\right )} x^{7} + \frac {2}{3} \, {\left (c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{4} + 8 \, b c d^{3} e + 6 \, b^{2} d^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{4} + 2 \, b^{2} d^{3} e\right )} x^{4} \]
1/9*c^2*e^4*x^9 + 1/3*b^2*d^4*x^3 + 1/4*(2*c^2*d*e^3 + b*c*e^4)*x^8 + 1/7* (6*c^2*d^2*e^2 + 8*b*c*d*e^3 + b^2*e^4)*x^7 + 2/3*(c^2*d^3*e + 3*b*c*d^2*e ^2 + b^2*d*e^3)*x^6 + 1/5*(c^2*d^4 + 8*b*c*d^3*e + 6*b^2*d^2*e^2)*x^5 + 1/ 2*(b*c*d^4 + 2*b^2*d^3*e)*x^4
Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.28 \[ \int (d+e x)^4 \left (b x+c x^2\right )^2 \, dx=\frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{2} \, c^{2} d e^{3} x^{8} + \frac {1}{4} \, b c e^{4} x^{8} + \frac {6}{7} \, c^{2} d^{2} e^{2} x^{7} + \frac {8}{7} \, b c d e^{3} x^{7} + \frac {1}{7} \, b^{2} e^{4} x^{7} + \frac {2}{3} \, c^{2} d^{3} e x^{6} + 2 \, b c d^{2} e^{2} x^{6} + \frac {2}{3} \, b^{2} d e^{3} x^{6} + \frac {1}{5} \, c^{2} d^{4} x^{5} + \frac {8}{5} \, b c d^{3} e x^{5} + \frac {6}{5} \, b^{2} d^{2} e^{2} x^{5} + \frac {1}{2} \, b c d^{4} x^{4} + b^{2} d^{3} e x^{4} + \frac {1}{3} \, b^{2} d^{4} x^{3} \]
1/9*c^2*e^4*x^9 + 1/2*c^2*d*e^3*x^8 + 1/4*b*c*e^4*x^8 + 6/7*c^2*d^2*e^2*x^ 7 + 8/7*b*c*d*e^3*x^7 + 1/7*b^2*e^4*x^7 + 2/3*c^2*d^3*e*x^6 + 2*b*c*d^2*e^ 2*x^6 + 2/3*b^2*d*e^3*x^6 + 1/5*c^2*d^4*x^5 + 8/5*b*c*d^3*e*x^5 + 6/5*b^2* d^2*e^2*x^5 + 1/2*b*c*d^4*x^4 + b^2*d^3*e*x^4 + 1/3*b^2*d^4*x^3
Time = 0.07 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.09 \[ \int (d+e x)^4 \left (b x+c x^2\right )^2 \, dx=x^5\,\left (\frac {6\,b^2\,d^2\,e^2}{5}+\frac {8\,b\,c\,d^3\,e}{5}+\frac {c^2\,d^4}{5}\right )+x^7\,\left (\frac {b^2\,e^4}{7}+\frac {8\,b\,c\,d\,e^3}{7}+\frac {6\,c^2\,d^2\,e^2}{7}\right )+\frac {b^2\,d^4\,x^3}{3}+\frac {c^2\,e^4\,x^9}{9}+\frac {b\,d^3\,x^4\,\left (2\,b\,e+c\,d\right )}{2}+\frac {c\,e^3\,x^8\,\left (b\,e+2\,c\,d\right )}{4}+\frac {2\,d\,e\,x^6\,\left (b^2\,e^2+3\,b\,c\,d\,e+c^2\,d^2\right )}{3} \]