Integrand size = 19, antiderivative size = 87 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} b^2 d^2 x^3+\frac {1}{2} b d (c d+b e) x^4+\frac {1}{5} \left (c^2 d^2+4 b c d e+b^2 e^2\right ) x^5+\frac {1}{3} c e (c d+b e) x^6+\frac {1}{7} c^2 e^2 x^7 \] Output:
1/3*b^2*d^2*x^3+1/2*b*d*(b*e+c*d)*x^4+1/5*(b^2*e^2+4*b*c*d*e+c^2*d^2)*x^5+ 1/3*c*e*(b*e+c*d)*x^6+1/7*c^2*e^2*x^7
Time = 0.01 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} b^2 d^2 x^3+\frac {1}{2} b d (c d+b e) x^4+\frac {1}{5} \left (c^2 d^2+4 b c d e+b^2 e^2\right ) x^5+\frac {1}{3} c e (c d+b e) x^6+\frac {1}{7} c^2 e^2 x^7 \] Input:
Integrate[(d + e*x)^2*(b*x + c*x^2)^2,x]
Output:
(b^2*d^2*x^3)/3 + (b*d*(c*d + b*e)*x^4)/2 + ((c^2*d^2 + 4*b*c*d*e + b^2*e^ 2)*x^5)/5 + (c*e*(c*d + b*e)*x^6)/3 + (c^2*e^2*x^7)/7
Time = 0.42 (sec) , antiderivative size = 87, 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.
\(\displaystyle \int \left (b x+c x^2\right )^2 (d+e x)^2 \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (x^4 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+b^2 d^2 x^2+2 c e x^5 (b e+c d)+2 b d x^3 (b e+c d)+c^2 e^2 x^6\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac {1}{3} b^2 d^2 x^3+\frac {1}{3} c e x^6 (b e+c d)+\frac {1}{2} b d x^4 (b e+c d)+\frac {1}{7} c^2 e^2 x^7\) |
Input:
Int[(d + e*x)^2*(b*x + c*x^2)^2,x]
Output:
(b^2*d^2*x^3)/3 + (b*d*(c*d + b*e)*x^4)/2 + ((c^2*d^2 + 4*b*c*d*e + b^2*e^ 2)*x^5)/5 + (c*e*(c*d + b*e)*x^6)/3 + (c^2*e^2*x^7)/7
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 = 0.33 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02
method | result | size |
norman | \(\frac {c^{2} e^{2} x^{7}}{7}+\left (\frac {1}{3} e^{2} b c +\frac {1}{3} d e \,c^{2}\right ) x^{6}+\left (\frac {1}{5} b^{2} e^{2}+\frac {4}{5} b c d e +\frac {1}{5} c^{2} d^{2}\right ) x^{5}+\left (\frac {1}{2} d e \,b^{2}+\frac {1}{2} b c \,d^{2}\right ) x^{4}+\frac {x^{3} b^{2} d^{2}}{3}\) | \(89\) |
default | \(\frac {c^{2} e^{2} x^{7}}{7}+\frac {\left (2 e^{2} b c +2 d e \,c^{2}\right ) x^{6}}{6}+\frac {\left (b^{2} e^{2}+4 b c d e +c^{2} d^{2}\right ) x^{5}}{5}+\frac {\left (2 d e \,b^{2}+2 b c \,d^{2}\right ) x^{4}}{4}+\frac {x^{3} b^{2} d^{2}}{3}\) | \(90\) |
gosper | \(\frac {x^{3} \left (30 c^{2} e^{2} x^{4}+70 x^{3} e^{2} b c +70 x^{3} d e \,c^{2}+42 x^{2} b^{2} e^{2}+168 x^{2} b c d e +42 d^{2} c^{2} x^{2}+105 x d e \,b^{2}+105 x b c \,d^{2}+70 b^{2} d^{2}\right )}{210}\) | \(93\) |
risch | \(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} x^{6} e^{2} b c +\frac {1}{3} d e \,c^{2} x^{6}+\frac {1}{5} x^{5} b^{2} e^{2}+\frac {4}{5} x^{5} b c d e +\frac {1}{5} x^{5} c^{2} d^{2}+\frac {1}{2} x^{4} d e \,b^{2}+\frac {1}{2} b c \,d^{2} x^{4}+\frac {1}{3} x^{3} b^{2} d^{2}\) | \(95\) |
parallelrisch | \(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} x^{6} e^{2} b c +\frac {1}{3} d e \,c^{2} x^{6}+\frac {1}{5} x^{5} b^{2} e^{2}+\frac {4}{5} x^{5} b c d e +\frac {1}{5} x^{5} c^{2} d^{2}+\frac {1}{2} x^{4} d e \,b^{2}+\frac {1}{2} b c \,d^{2} x^{4}+\frac {1}{3} x^{3} b^{2} d^{2}\) | \(95\) |
orering | \(\frac {x \left (30 c^{2} e^{2} x^{4}+70 x^{3} e^{2} b c +70 x^{3} d e \,c^{2}+42 x^{2} b^{2} e^{2}+168 x^{2} b c d e +42 d^{2} c^{2} x^{2}+105 x d e \,b^{2}+105 x b c \,d^{2}+70 b^{2} d^{2}\right ) \left (c \,x^{2}+b x \right )^{2}}{210 \left (c x +b \right )^{2}}\) | \(109\) |
Input:
int((e*x+d)^2*(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)
Output:
1/7*c^2*e^2*x^7+(1/3*e^2*b*c+1/3*d*e*c^2)*x^6+(1/5*b^2*e^2+4/5*b*c*d*e+1/5 *c^2*d^2)*x^5+(1/2*d*e*b^2+1/2*b*c*d^2)*x^4+1/3*x^3*b^2*d^2
Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, b^{2} d^{2} x^{3} + \frac {1}{3} \, {\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{2} + b^{2} d e\right )} x^{4} \] Input:
integrate((e*x+d)^2*(c*x^2+b*x)^2,x, algorithm="fricas")
Output:
1/7*c^2*e^2*x^7 + 1/3*b^2*d^2*x^3 + 1/3*(c^2*d*e + b*c*e^2)*x^6 + 1/5*(c^2 *d^2 + 4*b*c*d*e + b^2*e^2)*x^5 + 1/2*(b*c*d^2 + b^2*d*e)*x^4
Time = 0.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {b^{2} d^{2} x^{3}}{3} + \frac {c^{2} e^{2} x^{7}}{7} + x^{6} \left (\frac {b c e^{2}}{3} + \frac {c^{2} d e}{3}\right ) + x^{5} \left (\frac {b^{2} e^{2}}{5} + \frac {4 b c d e}{5} + \frac {c^{2} d^{2}}{5}\right ) + x^{4} \left (\frac {b^{2} d e}{2} + \frac {b c d^{2}}{2}\right ) \] Input:
integrate((e*x+d)**2*(c*x**2+b*x)**2,x)
Output:
b**2*d**2*x**3/3 + c**2*e**2*x**7/7 + x**6*(b*c*e**2/3 + c**2*d*e/3) + x** 5*(b**2*e**2/5 + 4*b*c*d*e/5 + c**2*d**2/5) + x**4*(b**2*d*e/2 + b*c*d**2/ 2)
Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, b^{2} d^{2} x^{3} + \frac {1}{3} \, {\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{2} + b^{2} d e\right )} x^{4} \] Input:
integrate((e*x+d)^2*(c*x^2+b*x)^2,x, algorithm="maxima")
Output:
1/7*c^2*e^2*x^7 + 1/3*b^2*d^2*x^3 + 1/3*(c^2*d*e + b*c*e^2)*x^6 + 1/5*(c^2 *d^2 + 4*b*c*d*e + b^2*e^2)*x^5 + 1/2*(b*c*d^2 + b^2*d*e)*x^4
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, c^{2} d e x^{6} + \frac {1}{3} \, b c e^{2} x^{6} + \frac {1}{5} \, c^{2} d^{2} x^{5} + \frac {4}{5} \, b c d e x^{5} + \frac {1}{5} \, b^{2} e^{2} x^{5} + \frac {1}{2} \, b c d^{2} x^{4} + \frac {1}{2} \, b^{2} d e x^{4} + \frac {1}{3} \, b^{2} d^{2} x^{3} \] Input:
integrate((e*x+d)^2*(c*x^2+b*x)^2,x, algorithm="giac")
Output:
1/7*c^2*e^2*x^7 + 1/3*c^2*d*e*x^6 + 1/3*b*c*e^2*x^6 + 1/5*c^2*d^2*x^5 + 4/ 5*b*c*d*e*x^5 + 1/5*b^2*e^2*x^5 + 1/2*b*c*d^2*x^4 + 1/2*b^2*d*e*x^4 + 1/3* b^2*d^2*x^3
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=x^5\,\left (\frac {b^2\,e^2}{5}+\frac {4\,b\,c\,d\,e}{5}+\frac {c^2\,d^2}{5}\right )+\frac {b^2\,d^2\,x^3}{3}+\frac {c^2\,e^2\,x^7}{7}+\frac {b\,d\,x^4\,\left (b\,e+c\,d\right )}{2}+\frac {c\,e\,x^6\,\left (b\,e+c\,d\right )}{3} \] Input:
int((b*x + c*x^2)^2*(d + e*x)^2,x)
Output:
x^5*((b^2*e^2)/5 + (c^2*d^2)/5 + (4*b*c*d*e)/5) + (b^2*d^2*x^3)/3 + (c^2*e ^2*x^7)/7 + (b*d*x^4*(b*e + c*d))/2 + (c*e*x^6*(b*e + c*d))/3
Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.06 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {x^{3} \left (30 c^{2} e^{2} x^{4}+70 b c \,e^{2} x^{3}+70 c^{2} d e \,x^{3}+42 b^{2} e^{2} x^{2}+168 b c d e \,x^{2}+42 c^{2} d^{2} x^{2}+105 b^{2} d e x +105 b c \,d^{2} x +70 b^{2} d^{2}\right )}{210} \] Input:
int((e*x+d)^2*(c*x^2+b*x)^2,x)
Output:
(x**3*(70*b**2*d**2 + 105*b**2*d*e*x + 42*b**2*e**2*x**2 + 105*b*c*d**2*x + 168*b*c*d*e*x**2 + 70*b*c*e**2*x**3 + 42*c**2*d**2*x**2 + 70*c**2*d*e*x* *3 + 30*c**2*e**2*x**4))/210