Integrand size = 24, antiderivative size = 162 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} A b^2 d^2 x^3+\frac {1}{4} b d (b B d+2 A c d+2 A b e) x^4+\frac {1}{5} \left (A c^2 d^2+b^2 e (2 B d+A e)+2 b c d (B d+2 A e)\right ) x^5+\frac {1}{6} \left (2 A c e (c d+b e)+B \left (c^2 d^2+4 b c d e+b^2 e^2\right )\right ) x^6+\frac {1}{7} c e (A c e+2 B (c d+b e)) x^7+\frac {1}{8} B c^2 e^2 x^8 \] Output:
1/3*A*b^2*d^2*x^3+1/4*b*d*(2*A*b*e+2*A*c*d+B*b*d)*x^4+1/5*(A*c^2*d^2+b^2*e *(A*e+2*B*d)+2*b*c*d*(2*A*e+B*d))*x^5+1/6*(2*A*c*e*(b*e+c*d)+B*(b^2*e^2+4* b*c*d*e+c^2*d^2))*x^6+1/7*c*e*(A*c*e+2*B*(b*e+c*d))*x^7+1/8*B*c^2*e^2*x^8
Time = 0.07 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} A b^2 d^2 x^3+\frac {1}{4} b d (b B d+2 A c d+2 A b e) x^4+\frac {1}{5} \left (A c^2 d^2+b^2 e (2 B d+A e)+2 b c d (B d+2 A e)\right ) x^5+\frac {1}{6} \left (2 A c e (c d+b e)+B \left (c^2 d^2+4 b c d e+b^2 e^2\right )\right ) x^6+\frac {1}{7} c e (A c e+2 B (c d+b e)) x^7+\frac {1}{8} B c^2 e^2 x^8 \] Input:
Integrate[(A + B*x)*(d + e*x)^2*(b*x + c*x^2)^2,x]
Output:
(A*b^2*d^2*x^3)/3 + (b*d*(b*B*d + 2*A*c*d + 2*A*b*e)*x^4)/4 + ((A*c^2*d^2 + b^2*e*(2*B*d + A*e) + 2*b*c*d*(B*d + 2*A*e))*x^5)/5 + ((2*A*c*e*(c*d + b *e) + B*(c^2*d^2 + 4*b*c*d*e + b^2*e^2))*x^6)/6 + (c*e*(A*c*e + 2*B*(c*d + b*e))*x^7)/7 + (B*c^2*e^2*x^8)/8
Time = 0.67 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1195, 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.
| \(\displaystyle \int (A+B x) \left (b x+c x^2\right )^2 (d+e x)^2 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (x^5 \left (2 A c e (b e+c d)+B \left (b^2 e^2+4 b c d e+c^2 d^2\right )\right )+x^4 \left (b^2 e (A e+2 B d)+2 b c d (2 A e+B d)+A c^2 d^2\right )+A b^2 d^2 x^2+c e x^6 (A c e+2 B (b e+c d))+b d x^3 (2 A b e+2 A c d+b B d)+B c^2 e^2 x^7\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} x^6 \left (2 A c e (b e+c d)+B \left (b^2 e^2+4 b c d e+c^2 d^2\right )\right )+\frac {1}{5} x^5 \left (b^2 e (A e+2 B d)+2 b c d (2 A e+B d)+A c^2 d^2\right )+\frac {1}{3} A b^2 d^2 x^3+\frac {1}{7} c e x^7 (A c e+2 B (b e+c d))+\frac {1}{4} b d x^4 (2 A b e+2 A c d+b B d)+\frac {1}{8} B c^2 e^2 x^8\) |
Input:
Int[(A + B*x)*(d + e*x)^2*(b*x + c*x^2)^2,x]
Output:
(A*b^2*d^2*x^3)/3 + (b*d*(b*B*d + 2*A*c*d + 2*A*b*e)*x^4)/4 + ((A*c^2*d^2 + b^2*e*(2*B*d + A*e) + 2*b*c*d*(B*d + 2*A*e))*x^5)/5 + ((2*A*c*e*(c*d + b *e) + B*(c^2*d^2 + 4*b*c*d*e + b^2*e^2))*x^6)/6 + (c*e*(A*c*e + 2*B*(c*d + b*e))*x^7)/7 + (B*c^2*e^2*x^8)/8
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.68 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.06
| method | result | size |
| default | \(\frac {B \,e^{2} c^{2} x^{8}}{8}+\frac {\left (\left (A \,e^{2}+2 B d e \right ) c^{2}+2 B \,e^{2} b c \right ) x^{7}}{7}+\frac {\left (\left (2 A d e +B \,d^{2}\right ) c^{2}+2 \left (A \,e^{2}+2 B d e \right ) b c +B \,e^{2} b^{2}\right ) x^{6}}{6}+\frac {\left (A \,c^{2} d^{2}+2 \left (2 A d e +B \,d^{2}\right ) b c +\left (A \,e^{2}+2 B d e \right ) b^{2}\right ) x^{5}}{5}+\frac {\left (2 A b c \,d^{2}+\left (2 A d e +B \,d^{2}\right ) b^{2}\right ) x^{4}}{4}+\frac {A \,b^{2} d^{2} x^{3}}{3}\) | \(172\) |
| norman | \(\frac {B \,e^{2} c^{2} x^{8}}{8}+\left (\frac {1}{7} A \,c^{2} e^{2}+\frac {2}{7} B \,e^{2} b c +\frac {2}{7} B \,c^{2} d e \right ) x^{7}+\left (\frac {1}{3} A b c \,e^{2}+\frac {1}{3} A \,c^{2} d e +\frac {1}{6} B \,e^{2} b^{2}+\frac {2}{3} B b c d e +\frac {1}{6} B \,c^{2} d^{2}\right ) x^{6}+\left (\frac {1}{5} A \,b^{2} e^{2}+\frac {4}{5} A b c d e +\frac {1}{5} A \,c^{2} d^{2}+\frac {2}{5} B \,b^{2} d e +\frac {2}{5} B b c \,d^{2}\right ) x^{5}+\left (\frac {1}{2} A \,b^{2} d e +\frac {1}{2} A b c \,d^{2}+\frac {1}{4} B \,b^{2} d^{2}\right ) x^{4}+\frac {A \,b^{2} d^{2} x^{3}}{3}\) | \(178\) |
| gosper | \(\frac {x^{3} \left (105 B \,e^{2} c^{2} x^{5}+120 x^{4} A \,c^{2} e^{2}+240 x^{4} B \,e^{2} b c +240 x^{4} B \,c^{2} d e +280 x^{3} A b c \,e^{2}+280 x^{3} A \,c^{2} d e +140 x^{3} B \,e^{2} b^{2}+560 x^{3} B b c d e +140 x^{3} B \,c^{2} d^{2}+168 x^{2} A \,b^{2} e^{2}+672 x^{2} A b c d e +168 x^{2} A \,c^{2} d^{2}+336 x^{2} B \,b^{2} d e +336 B b c \,d^{2} x^{2}+420 x A \,b^{2} d e +420 A b c \,d^{2} x +210 B \,b^{2} d^{2} x +280 A \,b^{2} d^{2}\right )}{840}\) | \(202\) |
| risch | \(\frac {1}{8} B \,e^{2} c^{2} x^{8}+\frac {1}{7} x^{7} A \,c^{2} e^{2}+\frac {2}{7} x^{7} B \,e^{2} b c +\frac {2}{7} x^{7} B \,c^{2} d e +\frac {1}{3} x^{6} A b c \,e^{2}+\frac {1}{3} x^{6} A \,c^{2} d e +\frac {1}{6} x^{6} B \,e^{2} b^{2}+\frac {2}{3} x^{6} B b c d e +\frac {1}{6} x^{6} B \,c^{2} d^{2}+\frac {1}{5} x^{5} A \,b^{2} e^{2}+\frac {4}{5} x^{5} A b c d e +\frac {1}{5} x^{5} A \,c^{2} d^{2}+\frac {2}{5} x^{5} B \,b^{2} d e +\frac {2}{5} x^{5} B b c \,d^{2}+\frac {1}{2} x^{4} A \,b^{2} d e +\frac {1}{2} x^{4} A b c \,d^{2}+\frac {1}{4} x^{4} B \,b^{2} d^{2}+\frac {1}{3} A \,b^{2} d^{2} x^{3}\) | \(206\) |
| parallelrisch | \(\frac {1}{8} B \,e^{2} c^{2} x^{8}+\frac {1}{7} x^{7} A \,c^{2} e^{2}+\frac {2}{7} x^{7} B \,e^{2} b c +\frac {2}{7} x^{7} B \,c^{2} d e +\frac {1}{3} x^{6} A b c \,e^{2}+\frac {1}{3} x^{6} A \,c^{2} d e +\frac {1}{6} x^{6} B \,e^{2} b^{2}+\frac {2}{3} x^{6} B b c d e +\frac {1}{6} x^{6} B \,c^{2} d^{2}+\frac {1}{5} x^{5} A \,b^{2} e^{2}+\frac {4}{5} x^{5} A b c d e +\frac {1}{5} x^{5} A \,c^{2} d^{2}+\frac {2}{5} x^{5} B \,b^{2} d e +\frac {2}{5} x^{5} B b c \,d^{2}+\frac {1}{2} x^{4} A \,b^{2} d e +\frac {1}{2} x^{4} A b c \,d^{2}+\frac {1}{4} x^{4} B \,b^{2} d^{2}+\frac {1}{3} A \,b^{2} d^{2} x^{3}\) | \(206\) |
| orering | \(\frac {x \left (105 B \,e^{2} c^{2} x^{5}+120 x^{4} A \,c^{2} e^{2}+240 x^{4} B \,e^{2} b c +240 x^{4} B \,c^{2} d e +280 x^{3} A b c \,e^{2}+280 x^{3} A \,c^{2} d e +140 x^{3} B \,e^{2} b^{2}+560 x^{3} B b c d e +140 x^{3} B \,c^{2} d^{2}+168 x^{2} A \,b^{2} e^{2}+672 x^{2} A b c d e +168 x^{2} A \,c^{2} d^{2}+336 x^{2} B \,b^{2} d e +336 B b c \,d^{2} x^{2}+420 x A \,b^{2} d e +420 A b c \,d^{2} x +210 B \,b^{2} d^{2} x +280 A \,b^{2} d^{2}\right ) \left (c \,x^{2}+b x \right )^{2}}{840 \left (c x +b \right )^{2}}\) | \(218\) |
Input:
int((B*x+A)*(e*x+d)^2*(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)
Output:
1/8*B*e^2*c^2*x^8+1/7*((A*e^2+2*B*d*e)*c^2+2*B*e^2*b*c)*x^7+1/6*((2*A*d*e+ B*d^2)*c^2+2*(A*e^2+2*B*d*e)*b*c+B*e^2*b^2)*x^6+1/5*(A*c^2*d^2+2*(2*A*d*e+ B*d^2)*b*c+(A*e^2+2*B*d*e)*b^2)*x^5+1/4*(2*A*b*c*d^2+(2*A*d*e+B*d^2)*b^2)* x^4+1/3*A*b^2*d^2*x^3
Time = 0.06 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.06 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{8} \, B c^{2} e^{2} x^{8} + \frac {1}{3} \, A b^{2} d^{2} x^{3} + \frac {1}{7} \, {\left (2 \, B c^{2} d e + {\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{2} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (A b^{2} e^{2} + {\left (2 \, B b c + A c^{2}\right )} d^{2} + 2 \, {\left (B b^{2} + 2 \, A b c\right )} d e\right )} x^{5} + \frac {1}{4} \, {\left (2 \, A b^{2} d e + {\left (B b^{2} + 2 \, A b c\right )} d^{2}\right )} x^{4} \] Input:
integrate((B*x+A)*(e*x+d)^2*(c*x^2+b*x)^2,x, algorithm="fricas")
Output:
1/8*B*c^2*e^2*x^8 + 1/3*A*b^2*d^2*x^3 + 1/7*(2*B*c^2*d*e + (2*B*b*c + A*c^ 2)*e^2)*x^7 + 1/6*(B*c^2*d^2 + 2*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c) *e^2)*x^6 + 1/5*(A*b^2*e^2 + (2*B*b*c + A*c^2)*d^2 + 2*(B*b^2 + 2*A*b*c)*d *e)*x^5 + 1/4*(2*A*b^2*d*e + (B*b^2 + 2*A*b*c)*d^2)*x^4
Time = 0.03 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.31 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {A b^{2} d^{2} x^{3}}{3} + \frac {B c^{2} e^{2} x^{8}}{8} + x^{7} \left (\frac {A c^{2} e^{2}}{7} + \frac {2 B b c e^{2}}{7} + \frac {2 B c^{2} d e}{7}\right ) + x^{6} \left (\frac {A b c e^{2}}{3} + \frac {A c^{2} d e}{3} + \frac {B b^{2} e^{2}}{6} + \frac {2 B b c d e}{3} + \frac {B c^{2} d^{2}}{6}\right ) + x^{5} \left (\frac {A b^{2} e^{2}}{5} + \frac {4 A b c d e}{5} + \frac {A c^{2} d^{2}}{5} + \frac {2 B b^{2} d e}{5} + \frac {2 B b c d^{2}}{5}\right ) + x^{4} \left (\frac {A b^{2} d e}{2} + \frac {A b c d^{2}}{2} + \frac {B b^{2} d^{2}}{4}\right ) \] Input:
integrate((B*x+A)*(e*x+d)**2*(c*x**2+b*x)**2,x)
Output:
A*b**2*d**2*x**3/3 + B*c**2*e**2*x**8/8 + x**7*(A*c**2*e**2/7 + 2*B*b*c*e* *2/7 + 2*B*c**2*d*e/7) + x**6*(A*b*c*e**2/3 + A*c**2*d*e/3 + B*b**2*e**2/6 + 2*B*b*c*d*e/3 + B*c**2*d**2/6) + x**5*(A*b**2*e**2/5 + 4*A*b*c*d*e/5 + A*c**2*d**2/5 + 2*B*b**2*d*e/5 + 2*B*b*c*d**2/5) + x**4*(A*b**2*d*e/2 + A* b*c*d**2/2 + B*b**2*d**2/4)
Time = 0.03 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.06 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{8} \, B c^{2} e^{2} x^{8} + \frac {1}{3} \, A b^{2} d^{2} x^{3} + \frac {1}{7} \, {\left (2 \, B c^{2} d e + {\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{2} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (A b^{2} e^{2} + {\left (2 \, B b c + A c^{2}\right )} d^{2} + 2 \, {\left (B b^{2} + 2 \, A b c\right )} d e\right )} x^{5} + \frac {1}{4} \, {\left (2 \, A b^{2} d e + {\left (B b^{2} + 2 \, A b c\right )} d^{2}\right )} x^{4} \] Input:
integrate((B*x+A)*(e*x+d)^2*(c*x^2+b*x)^2,x, algorithm="maxima")
Output:
1/8*B*c^2*e^2*x^8 + 1/3*A*b^2*d^2*x^3 + 1/7*(2*B*c^2*d*e + (2*B*b*c + A*c^ 2)*e^2)*x^7 + 1/6*(B*c^2*d^2 + 2*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c) *e^2)*x^6 + 1/5*(A*b^2*e^2 + (2*B*b*c + A*c^2)*d^2 + 2*(B*b^2 + 2*A*b*c)*d *e)*x^5 + 1/4*(2*A*b^2*d*e + (B*b^2 + 2*A*b*c)*d^2)*x^4
Time = 0.23 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.27 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{8} \, B c^{2} e^{2} x^{8} + \frac {2}{7} \, B c^{2} d e x^{7} + \frac {2}{7} \, B b c e^{2} x^{7} + \frac {1}{7} \, A c^{2} e^{2} x^{7} + \frac {1}{6} \, B c^{2} d^{2} x^{6} + \frac {2}{3} \, B b c d e x^{6} + \frac {1}{3} \, A c^{2} d e x^{6} + \frac {1}{6} \, B b^{2} e^{2} x^{6} + \frac {1}{3} \, A b c e^{2} x^{6} + \frac {2}{5} \, B b c d^{2} x^{5} + \frac {1}{5} \, A c^{2} d^{2} x^{5} + \frac {2}{5} \, B b^{2} d e x^{5} + \frac {4}{5} \, A b c d e x^{5} + \frac {1}{5} \, A b^{2} e^{2} x^{5} + \frac {1}{4} \, B b^{2} d^{2} x^{4} + \frac {1}{2} \, A b c d^{2} x^{4} + \frac {1}{2} \, A b^{2} d e x^{4} + \frac {1}{3} \, A b^{2} d^{2} x^{3} \] Input:
integrate((B*x+A)*(e*x+d)^2*(c*x^2+b*x)^2,x, algorithm="giac")
Output:
1/8*B*c^2*e^2*x^8 + 2/7*B*c^2*d*e*x^7 + 2/7*B*b*c*e^2*x^7 + 1/7*A*c^2*e^2* x^7 + 1/6*B*c^2*d^2*x^6 + 2/3*B*b*c*d*e*x^6 + 1/3*A*c^2*d*e*x^6 + 1/6*B*b^ 2*e^2*x^6 + 1/3*A*b*c*e^2*x^6 + 2/5*B*b*c*d^2*x^5 + 1/5*A*c^2*d^2*x^5 + 2/ 5*B*b^2*d*e*x^5 + 4/5*A*b*c*d*e*x^5 + 1/5*A*b^2*e^2*x^5 + 1/4*B*b^2*d^2*x^ 4 + 1/2*A*b*c*d^2*x^4 + 1/2*A*b^2*d*e*x^4 + 1/3*A*b^2*d^2*x^3
Time = 0.06 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.99 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=x^5\,\left (\frac {2\,B\,b^2\,d\,e}{5}+\frac {A\,b^2\,e^2}{5}+\frac {2\,B\,b\,c\,d^2}{5}+\frac {4\,A\,b\,c\,d\,e}{5}+\frac {A\,c^2\,d^2}{5}\right )+x^6\,\left (\frac {B\,b^2\,e^2}{6}+\frac {2\,B\,b\,c\,d\,e}{3}+\frac {A\,b\,c\,e^2}{3}+\frac {B\,c^2\,d^2}{6}+\frac {A\,c^2\,d\,e}{3}\right )+\frac {b\,d\,x^4\,\left (2\,A\,b\,e+2\,A\,c\,d+B\,b\,d\right )}{4}+\frac {c\,e\,x^7\,\left (A\,c\,e+2\,B\,b\,e+2\,B\,c\,d\right )}{7}+\frac {A\,b^2\,d^2\,x^3}{3}+\frac {B\,c^2\,e^2\,x^8}{8} \] Input:
int((b*x + c*x^2)^2*(A + B*x)*(d + e*x)^2,x)
Output:
x^5*((A*b^2*e^2)/5 + (A*c^2*d^2)/5 + (2*B*b*c*d^2)/5 + (2*B*b^2*d*e)/5 + ( 4*A*b*c*d*e)/5) + x^6*((B*b^2*e^2)/6 + (B*c^2*d^2)/6 + (A*b*c*e^2)/3 + (A* c^2*d*e)/3 + (2*B*b*c*d*e)/3) + (b*d*x^4*(2*A*b*e + 2*A*c*d + B*b*d))/4 + (c*e*x^7*(A*c*e + 2*B*b*e + 2*B*c*d))/7 + (A*b^2*d^2*x^3)/3 + (B*c^2*e^2*x ^8)/8
Time = 0.25 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.24 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {x^{3} \left (105 b \,c^{2} e^{2} x^{5}+120 a \,c^{2} e^{2} x^{4}+240 b^{2} c \,e^{2} x^{4}+240 b \,c^{2} d e \,x^{4}+280 a b c \,e^{2} x^{3}+280 a \,c^{2} d e \,x^{3}+140 b^{3} e^{2} x^{3}+560 b^{2} c d e \,x^{3}+140 b \,c^{2} d^{2} x^{3}+168 a \,b^{2} e^{2} x^{2}+672 a b c d e \,x^{2}+168 a \,c^{2} d^{2} x^{2}+336 b^{3} d e \,x^{2}+336 b^{2} c \,d^{2} x^{2}+420 a \,b^{2} d e x +420 a b c \,d^{2} x +210 b^{3} d^{2} x +280 a \,b^{2} d^{2}\right )}{840} \] Input:
int((B*x+A)*(e*x+d)^2*(c*x^2+b*x)^2,x)
Output:
(x**3*(280*a*b**2*d**2 + 420*a*b**2*d*e*x + 168*a*b**2*e**2*x**2 + 420*a*b *c*d**2*x + 672*a*b*c*d*e*x**2 + 280*a*b*c*e**2*x**3 + 168*a*c**2*d**2*x** 2 + 280*a*c**2*d*e*x**3 + 120*a*c**2*e**2*x**4 + 210*b**3*d**2*x + 336*b** 3*d*e*x**2 + 140*b**3*e**2*x**3 + 336*b**2*c*d**2*x**2 + 560*b**2*c*d*e*x* *3 + 240*b**2*c*e**2*x**4 + 140*b*c**2*d**2*x**3 + 240*b*c**2*d*e*x**4 + 1 05*b*c**2*e**2*x**5))/840