Integrand size = 19, antiderivative size = 127 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} b^2 d^3 x^3+\frac {1}{4} b d^2 (2 c d+3 b e) x^4+\frac {1}{5} d \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^5+\frac {1}{6} e \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} c e^2 (3 c d+2 b e) x^7+\frac {1}{8} c^2 e^3 x^8 \] Output:
1/3*b^2*d^3*x^3+1/4*b*d^2*(3*b*e+2*c*d)*x^4+1/5*d*(3*b^2*e^2+6*b*c*d*e+c^2 *d^2)*x^5+1/6*e*(b^2*e^2+6*b*c*d*e+3*c^2*d^2)*x^6+1/7*c*e^2*(2*b*e+3*c*d)* x^7+1/8*c^2*e^3*x^8
Time = 0.01 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} b^2 d^3 x^3+\frac {1}{4} b d^2 (2 c d+3 b e) x^4+\frac {1}{5} d \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^5+\frac {1}{6} e \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} c e^2 (3 c d+2 b e) x^7+\frac {1}{8} c^2 e^3 x^8 \] Input:
Integrate[(d + e*x)^3*(b*x + c*x^2)^2,x]
Output:
(b^2*d^3*x^3)/3 + (b*d^2*(2*c*d + 3*b*e)*x^4)/4 + (d*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2)*x^5)/5 + (e*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2)*x^6)/6 + (c*e^2 *(3*c*d + 2*b*e)*x^7)/7 + (c^2*e^3*x^8)/8
Time = 0.49 (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.
\(\displaystyle \int \left (b x+c x^2\right )^2 (d+e x)^3 \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (e x^5 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+d x^4 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+b^2 d^3 x^2+b d^2 x^3 (3 b e+2 c d)+c e^2 x^6 (2 b e+3 c d)+c^2 e^3 x^7\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{6} e x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac {1}{5} d x^5 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac {1}{3} b^2 d^3 x^3+\frac {1}{4} b d^2 x^4 (3 b e+2 c d)+\frac {1}{7} c e^2 x^7 (2 b e+3 c d)+\frac {1}{8} c^2 e^3 x^8\) |
Input:
Int[(d + e*x)^3*(b*x + c*x^2)^2,x]
Output:
(b^2*d^3*x^3)/3 + (b*d^2*(2*c*d + 3*b*e)*x^4)/4 + (d*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2)*x^5)/5 + (e*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2)*x^6)/6 + (c*e^2 *(3*c*d + 2*b*e)*x^7)/7 + (c^2*e^3*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 = 0.34 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.98
method | result | size |
norman | \(\frac {e^{3} c^{2} x^{8}}{8}+\left (\frac {2}{7} e^{3} b c +\frac {3}{7} d \,e^{2} c^{2}\right ) x^{7}+\left (\frac {1}{6} e^{3} b^{2}+d \,e^{2} b c +\frac {1}{2} d^{2} e \,c^{2}\right ) x^{6}+\left (\frac {3}{5} d \,e^{2} b^{2}+\frac {6}{5} d^{2} e b c +\frac {1}{5} c^{2} d^{3}\right ) x^{5}+\left (\frac {3}{4} d^{2} e \,b^{2}+\frac {1}{2} b c \,d^{3}\right ) x^{4}+\frac {b^{2} d^{3} x^{3}}{3}\) | \(125\) |
default | \(\frac {e^{3} c^{2} x^{8}}{8}+\frac {\left (2 e^{3} b c +3 d \,e^{2} c^{2}\right ) x^{7}}{7}+\frac {\left (e^{3} b^{2}+6 d \,e^{2} b c +3 d^{2} e \,c^{2}\right ) x^{6}}{6}+\frac {\left (3 d \,e^{2} b^{2}+6 d^{2} e b c +c^{2} d^{3}\right ) x^{5}}{5}+\frac {\left (3 d^{2} e \,b^{2}+2 b c \,d^{3}\right ) x^{4}}{4}+\frac {b^{2} d^{3} x^{3}}{3}\) | \(128\) |
gosper | \(\frac {x^{3} \left (105 e^{3} c^{2} x^{5}+240 x^{4} e^{3} b c +360 x^{4} d \,e^{2} c^{2}+140 x^{3} e^{3} b^{2}+840 x^{3} d \,e^{2} b c +420 x^{3} d^{2} e \,c^{2}+504 x^{2} d \,e^{2} b^{2}+1008 x^{2} d^{2} e b c +168 c^{2} d^{3} x^{2}+630 x \,d^{2} e \,b^{2}+420 b c \,d^{3} x +280 b^{2} d^{3}\right )}{840}\) | \(134\) |
risch | \(\frac {1}{8} e^{3} c^{2} x^{8}+\frac {2}{7} x^{7} e^{3} b c +\frac {3}{7} d \,e^{2} c^{2} x^{7}+\frac {1}{6} x^{6} e^{3} b^{2}+x^{6} d \,e^{2} b c +\frac {1}{2} x^{6} d^{2} e \,c^{2}+\frac {3}{5} x^{5} d \,e^{2} b^{2}+\frac {6}{5} x^{5} d^{2} e b c +\frac {1}{5} x^{5} c^{2} d^{3}+\frac {3}{4} x^{4} d^{2} e \,b^{2}+\frac {1}{2} c b \,d^{3} x^{4}+\frac {1}{3} b^{2} d^{3} x^{3}\) | \(135\) |
parallelrisch | \(\frac {1}{8} e^{3} c^{2} x^{8}+\frac {2}{7} x^{7} e^{3} b c +\frac {3}{7} d \,e^{2} c^{2} x^{7}+\frac {1}{6} x^{6} e^{3} b^{2}+x^{6} d \,e^{2} b c +\frac {1}{2} x^{6} d^{2} e \,c^{2}+\frac {3}{5} x^{5} d \,e^{2} b^{2}+\frac {6}{5} x^{5} d^{2} e b c +\frac {1}{5} x^{5} c^{2} d^{3}+\frac {3}{4} x^{4} d^{2} e \,b^{2}+\frac {1}{2} c b \,d^{3} x^{4}+\frac {1}{3} b^{2} d^{3} x^{3}\) | \(135\) |
orering | \(\frac {x \left (105 e^{3} c^{2} x^{5}+240 x^{4} e^{3} b c +360 x^{4} d \,e^{2} c^{2}+140 x^{3} e^{3} b^{2}+840 x^{3} d \,e^{2} b c +420 x^{3} d^{2} e \,c^{2}+504 x^{2} d \,e^{2} b^{2}+1008 x^{2} d^{2} e b c +168 c^{2} d^{3} x^{2}+630 x \,d^{2} e \,b^{2}+420 b c \,d^{3} x +280 b^{2} d^{3}\right ) \left (c \,x^{2}+b x \right )^{2}}{840 \left (c x +b \right )^{2}}\) | \(150\) |
Input:
int((e*x+d)^3*(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)
Output:
1/8*e^3*c^2*x^8+(2/7*e^3*b*c+3/7*d*e^2*c^2)*x^7+(1/6*e^3*b^2+d*e^2*b*c+1/2 *d^2*e*c^2)*x^6+(3/5*d*e^2*b^2+6/5*d^2*e*b*c+1/5*c^2*d^3)*x^5+(3/4*d^2*e*b ^2+1/2*b*c*d^3)*x^4+1/3*b^2*d^3*x^3
Time = 0.07 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{8} \, c^{2} e^{3} x^{8} + \frac {1}{3} \, b^{2} d^{3} x^{3} + \frac {1}{7} \, {\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + b^{2} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{3} + 6 \, b c d^{2} e + 3 \, b^{2} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d^{3} + 3 \, b^{2} d^{2} e\right )} x^{4} \] Input:
integrate((e*x+d)^3*(c*x^2+b*x)^2,x, algorithm="fricas")
Output:
1/8*c^2*e^3*x^8 + 1/3*b^2*d^3*x^3 + 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*e^3)*x^6 + 1/5*(c^2*d^3 + 6*b*c*d^2*e + 3*b^2*d*e^2)*x^5 + 1/4*(2*b*c*d^3 + 3*b^2*d^2*e)*x^4
Time = 0.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.09 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {b^{2} d^{3} x^{3}}{3} + \frac {c^{2} e^{3} x^{8}}{8} + x^{7} \cdot \left (\frac {2 b c e^{3}}{7} + \frac {3 c^{2} d e^{2}}{7}\right ) + x^{6} \left (\frac {b^{2} e^{3}}{6} + b c d e^{2} + \frac {c^{2} d^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {3 b^{2} d e^{2}}{5} + \frac {6 b c d^{2} e}{5} + \frac {c^{2} d^{3}}{5}\right ) + x^{4} \cdot \left (\frac {3 b^{2} d^{2} e}{4} + \frac {b c d^{3}}{2}\right ) \] Input:
integrate((e*x+d)**3*(c*x**2+b*x)**2,x)
Output:
b**2*d**3*x**3/3 + c**2*e**3*x**8/8 + x**7*(2*b*c*e**3/7 + 3*c**2*d*e**2/7 ) + x**6*(b**2*e**3/6 + b*c*d*e**2 + c**2*d**2*e/2) + x**5*(3*b**2*d*e**2/ 5 + 6*b*c*d**2*e/5 + c**2*d**3/5) + x**4*(3*b**2*d**2*e/4 + b*c*d**3/2)
Time = 0.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{8} \, c^{2} e^{3} x^{8} + \frac {1}{3} \, b^{2} d^{3} x^{3} + \frac {1}{7} \, {\left (3 \, c^{2} d e^{2} + 2 \, b c e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, c^{2} d^{2} e + 6 \, b c d e^{2} + b^{2} e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{3} + 6 \, b c d^{2} e + 3 \, b^{2} d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d^{3} + 3 \, b^{2} d^{2} e\right )} x^{4} \] Input:
integrate((e*x+d)^3*(c*x^2+b*x)^2,x, algorithm="maxima")
Output:
1/8*c^2*e^3*x^8 + 1/3*b^2*d^3*x^3 + 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*e^3)*x^6 + 1/5*(c^2*d^3 + 6*b*c*d^2*e + 3*b^2*d*e^2)*x^5 + 1/4*(2*b*c*d^3 + 3*b^2*d^2*e)*x^4
Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {1}{8} \, c^{2} e^{3} x^{8} + \frac {3}{7} \, c^{2} d e^{2} x^{7} + \frac {2}{7} \, b c e^{3} x^{7} + \frac {1}{2} \, c^{2} d^{2} e x^{6} + b c d e^{2} x^{6} + \frac {1}{6} \, b^{2} e^{3} x^{6} + \frac {1}{5} \, c^{2} d^{3} x^{5} + \frac {6}{5} \, b c d^{2} e x^{5} + \frac {3}{5} \, b^{2} d e^{2} x^{5} + \frac {1}{2} \, b c d^{3} x^{4} + \frac {3}{4} \, b^{2} d^{2} e x^{4} + \frac {1}{3} \, b^{2} d^{3} x^{3} \] Input:
integrate((e*x+d)^3*(c*x^2+b*x)^2,x, algorithm="giac")
Output:
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 + 1/5*c^2*d^3*x^5 + 6/5*b*c*d^2*e*x^5 + 3/5*b^2*d*e^2*x^5 + 1/2*b*c*d^3*x^4 + 3/4*b^2*d^2*e*x^4 + 1/3*b^2*d^3*x^3
Time = 9.01 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.93 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=x^5\,\left (\frac {3\,b^2\,d\,e^2}{5}+\frac {6\,b\,c\,d^2\,e}{5}+\frac {c^2\,d^3}{5}\right )+x^6\,\left (\frac {b^2\,e^3}{6}+b\,c\,d\,e^2+\frac {c^2\,d^2\,e}{2}\right )+\frac {b^2\,d^3\,x^3}{3}+\frac {c^2\,e^3\,x^8}{8}+\frac {b\,d^2\,x^4\,\left (3\,b\,e+2\,c\,d\right )}{4}+\frac {c\,e^2\,x^7\,\left (2\,b\,e+3\,c\,d\right )}{7} \] Input:
int((b*x + c*x^2)^2*(d + e*x)^3,x)
Output:
x^5*((c^2*d^3)/5 + (3*b^2*d*e^2)/5 + (6*b*c*d^2*e)/5) + x^6*((b^2*e^3)/6 + (c^2*d^2*e)/2 + b*c*d*e^2) + (b^2*d^3*x^3)/3 + (c^2*e^3*x^8)/8 + (b*d^2*x ^4*(3*b*e + 2*c*d))/4 + (c*e^2*x^7*(2*b*e + 3*c*d))/7
Time = 0.21 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.05 \[ \int (d+e x)^3 \left (b x+c x^2\right )^2 \, dx=\frac {x^{3} \left (105 c^{2} e^{3} x^{5}+240 b c \,e^{3} x^{4}+360 c^{2} d \,e^{2} x^{4}+140 b^{2} e^{3} x^{3}+840 b c d \,e^{2} x^{3}+420 c^{2} d^{2} e \,x^{3}+504 b^{2} d \,e^{2} x^{2}+1008 b c \,d^{2} e \,x^{2}+168 c^{2} d^{3} x^{2}+630 b^{2} d^{2} e x +420 b c \,d^{3} x +280 b^{2} d^{3}\right )}{840} \] Input:
int((e*x+d)^3*(c*x^2+b*x)^2,x)
Output:
(x**3*(280*b**2*d**3 + 630*b**2*d**2*e*x + 504*b**2*d*e**2*x**2 + 140*b**2 *e**3*x**3 + 420*b*c*d**3*x + 1008*b*c*d**2*e*x**2 + 840*b*c*d*e**2*x**3 + 240*b*c*e**3*x**4 + 168*c**2*d**3*x**2 + 420*c**2*d**2*e*x**3 + 360*c**2* d*e**2*x**4 + 105*c**2*e**3*x**5))/840