Integrand size = 22, antiderivative size = 99 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right ) \, dx=\frac {1}{2} A b d^2 x^2+\frac {1}{3} d (b B d+A c d+2 A b e) x^3+\frac {1}{4} (A e (2 c d+b e)+B d (c d+2 b e)) x^4+\frac {1}{5} e (2 B c d+b B e+A c e) x^5+\frac {1}{6} B c e^2 x^6 \] Output:
1/2*A*b*d^2*x^2+1/3*d*(2*A*b*e+A*c*d+B*b*d)*x^3+1/4*(A*e*(b*e+2*c*d)+B*d*( 2*b*e+c*d))*x^4+1/5*e*(A*c*e+B*b*e+2*B*c*d)*x^5+1/6*B*c*e^2*x^6
Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.92 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right ) \, dx=\frac {1}{60} x^2 \left (30 A b d^2+20 d (b B d+A c d+2 A b e) x+15 (A e (2 c d+b e)+B d (c d+2 b e)) x^2+12 e (2 B c d+b B e+A c e) x^3+10 B c e^2 x^4\right ) \] Input:
Integrate[(A + B*x)*(d + e*x)^2*(b*x + c*x^2),x]
Output:
(x^2*(30*A*b*d^2 + 20*d*(b*B*d + A*c*d + 2*A*b*e)*x + 15*(A*e*(2*c*d + b*e ) + B*d*(c*d + 2*b*e))*x^2 + 12*e*(2*B*c*d + b*B*e + A*c*e)*x^3 + 10*B*c*e ^2*x^4))/60
Time = 0.46 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
\(\displaystyle \int (A+B x) \left (b x+c x^2\right ) (d+e x)^2 \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (e x^4 (A c e+b B e+2 B c d)+x^3 (A e (b e+2 c d)+B d (2 b e+c d))+d x^2 (2 A b e+A c d+b B d)+A b d^2 x+B c e^2 x^5\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} e x^5 (A c e+b B e+2 B c d)+\frac {1}{4} x^4 (A e (b e+2 c d)+B d (2 b e+c d))+\frac {1}{3} d x^3 (2 A b e+A c d+b B d)+\frac {1}{2} A b d^2 x^2+\frac {1}{6} B c e^2 x^6\) |
Input:
Int[(A + B*x)*(d + e*x)^2*(b*x + c*x^2),x]
Output:
(A*b*d^2*x^2)/2 + (d*(b*B*d + A*c*d + 2*A*b*e)*x^3)/3 + ((A*e*(2*c*d + b*e ) + B*d*(c*d + 2*b*e))*x^4)/4 + (e*(2*B*c*d + b*B*e + A*c*e)*x^5)/5 + (B*c *e^2*x^6)/6
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 = 0.55 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.04
method | result | size |
norman | \(\frac {B \,e^{2} c \,x^{6}}{6}+\left (\frac {1}{5} A c \,e^{2}+\frac {1}{5} B \,e^{2} b +\frac {2}{5} B c d e \right ) x^{5}+\left (\frac {1}{4} A b \,e^{2}+\frac {1}{2} A c d e +\frac {1}{2} B b d e +\frac {1}{4} B c \,d^{2}\right ) x^{4}+\left (\frac {2}{3} A b d e +\frac {1}{3} A c \,d^{2}+\frac {1}{3} B b \,d^{2}\right ) x^{3}+\frac {A b \,d^{2} x^{2}}{2}\) | \(103\) |
default | \(\frac {B \,e^{2} c \,x^{6}}{6}+\frac {\left (\left (A \,e^{2}+2 B d e \right ) c +B \,e^{2} b \right ) x^{5}}{5}+\frac {\left (\left (2 A d e +B \,d^{2}\right ) c +\left (A \,e^{2}+2 B d e \right ) b \right ) x^{4}}{4}+\frac {\left (A c \,d^{2}+\left (2 A d e +B \,d^{2}\right ) b \right ) x^{3}}{3}+\frac {A b \,d^{2} x^{2}}{2}\) | \(104\) |
gosper | \(\frac {x^{2} \left (10 B \,e^{2} c \,x^{4}+12 x^{3} A c \,e^{2}+12 x^{3} B \,e^{2} b +24 x^{3} B c d e +15 x^{2} A b \,e^{2}+30 x^{2} A c d e +30 x^{2} B b d e +15 B c \,d^{2} x^{2}+40 x A b d e +20 A c \,d^{2} x +20 B b \,d^{2} x +30 A b \,d^{2}\right )}{60}\) | \(114\) |
risch | \(\frac {1}{6} B \,e^{2} c \,x^{6}+\frac {1}{5} x^{5} A c \,e^{2}+\frac {1}{5} x^{5} B \,e^{2} b +\frac {2}{5} x^{5} B c d e +\frac {1}{4} x^{4} A b \,e^{2}+\frac {1}{2} x^{4} A c d e +\frac {1}{2} x^{4} B b d e +\frac {1}{4} x^{4} B c \,d^{2}+\frac {2}{3} x^{3} A b d e +\frac {1}{3} x^{3} A c \,d^{2}+\frac {1}{3} x^{3} B b \,d^{2}+\frac {1}{2} A b \,d^{2} x^{2}\) | \(118\) |
parallelrisch | \(\frac {1}{6} B \,e^{2} c \,x^{6}+\frac {1}{5} x^{5} A c \,e^{2}+\frac {1}{5} x^{5} B \,e^{2} b +\frac {2}{5} x^{5} B c d e +\frac {1}{4} x^{4} A b \,e^{2}+\frac {1}{2} x^{4} A c d e +\frac {1}{2} x^{4} B b d e +\frac {1}{4} x^{4} B c \,d^{2}+\frac {2}{3} x^{3} A b d e +\frac {1}{3} x^{3} A c \,d^{2}+\frac {1}{3} x^{3} B b \,d^{2}+\frac {1}{2} A b \,d^{2} x^{2}\) | \(118\) |
orering | \(\frac {x \left (10 B \,e^{2} c \,x^{4}+12 x^{3} A c \,e^{2}+12 x^{3} B \,e^{2} b +24 x^{3} B c d e +15 x^{2} A b \,e^{2}+30 x^{2} A c d e +30 x^{2} B b d e +15 B c \,d^{2} x^{2}+40 x A b d e +20 A c \,d^{2} x +20 B b \,d^{2} x +30 A b \,d^{2}\right ) \left (c \,x^{2}+b x \right )}{60 c x +60 b}\) | \(128\) |
Input:
int((B*x+A)*(e*x+d)^2*(c*x^2+b*x),x,method=_RETURNVERBOSE)
Output:
1/6*B*e^2*c*x^6+(1/5*A*c*e^2+1/5*B*e^2*b+2/5*B*c*d*e)*x^5+(1/4*A*b*e^2+1/2 *A*c*d*e+1/2*B*b*d*e+1/4*B*c*d^2)*x^4+(2/3*A*b*d*e+1/3*A*c*d^2+1/3*B*b*d^2 )*x^3+1/2*A*b*d^2*x^2
Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.97 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right ) \, dx=\frac {1}{6} \, B c e^{2} x^{6} + \frac {1}{2} \, A b d^{2} x^{2} + \frac {1}{5} \, {\left (2 \, B c d e + {\left (B b + A c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{2} + A b e^{2} + 2 \, {\left (B b + A c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (2 \, A b d e + {\left (B b + A c\right )} d^{2}\right )} x^{3} \] Input:
integrate((B*x+A)*(e*x+d)^2*(c*x^2+b*x),x, algorithm="fricas")
Output:
1/6*B*c*e^2*x^6 + 1/2*A*b*d^2*x^2 + 1/5*(2*B*c*d*e + (B*b + A*c)*e^2)*x^5 + 1/4*(B*c*d^2 + A*b*e^2 + 2*(B*b + A*c)*d*e)*x^4 + 1/3*(2*A*b*d*e + (B*b + A*c)*d^2)*x^3
Time = 0.02 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.22 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right ) \, dx=\frac {A b d^{2} x^{2}}{2} + \frac {B c e^{2} x^{6}}{6} + x^{5} \left (\frac {A c e^{2}}{5} + \frac {B b e^{2}}{5} + \frac {2 B c d e}{5}\right ) + x^{4} \left (\frac {A b e^{2}}{4} + \frac {A c d e}{2} + \frac {B b d e}{2} + \frac {B c d^{2}}{4}\right ) + x^{3} \cdot \left (\frac {2 A b d e}{3} + \frac {A c d^{2}}{3} + \frac {B b d^{2}}{3}\right ) \] Input:
integrate((B*x+A)*(e*x+d)**2*(c*x**2+b*x),x)
Output:
A*b*d**2*x**2/2 + B*c*e**2*x**6/6 + x**5*(A*c*e**2/5 + B*b*e**2/5 + 2*B*c* d*e/5) + x**4*(A*b*e**2/4 + A*c*d*e/2 + B*b*d*e/2 + B*c*d**2/4) + x**3*(2* A*b*d*e/3 + A*c*d**2/3 + B*b*d**2/3)
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.97 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right ) \, dx=\frac {1}{6} \, B c e^{2} x^{6} + \frac {1}{2} \, A b d^{2} x^{2} + \frac {1}{5} \, {\left (2 \, B c d e + {\left (B b + A c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{2} + A b e^{2} + 2 \, {\left (B b + A c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (2 \, A b d e + {\left (B b + A c\right )} d^{2}\right )} x^{3} \] Input:
integrate((B*x+A)*(e*x+d)^2*(c*x^2+b*x),x, algorithm="maxima")
Output:
1/6*B*c*e^2*x^6 + 1/2*A*b*d^2*x^2 + 1/5*(2*B*c*d*e + (B*b + A*c)*e^2)*x^5 + 1/4*(B*c*d^2 + A*b*e^2 + 2*(B*b + A*c)*d*e)*x^4 + 1/3*(2*A*b*d*e + (B*b + A*c)*d^2)*x^3
Time = 0.24 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.18 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right ) \, dx=\frac {1}{6} \, B c e^{2} x^{6} + \frac {2}{5} \, B c d e x^{5} + \frac {1}{5} \, B b e^{2} x^{5} + \frac {1}{5} \, A c e^{2} x^{5} + \frac {1}{4} \, B c d^{2} x^{4} + \frac {1}{2} \, B b d e x^{4} + \frac {1}{2} \, A c d e x^{4} + \frac {1}{4} \, A b e^{2} x^{4} + \frac {1}{3} \, B b d^{2} x^{3} + \frac {1}{3} \, A c d^{2} x^{3} + \frac {2}{3} \, A b d e x^{3} + \frac {1}{2} \, A b d^{2} x^{2} \] Input:
integrate((B*x+A)*(e*x+d)^2*(c*x^2+b*x),x, algorithm="giac")
Output:
1/6*B*c*e^2*x^6 + 2/5*B*c*d*e*x^5 + 1/5*B*b*e^2*x^5 + 1/5*A*c*e^2*x^5 + 1/ 4*B*c*d^2*x^4 + 1/2*B*b*d*e*x^4 + 1/2*A*c*d*e*x^4 + 1/4*A*b*e^2*x^4 + 1/3* B*b*d^2*x^3 + 1/3*A*c*d^2*x^3 + 2/3*A*b*d*e*x^3 + 1/2*A*b*d^2*x^2
Time = 10.90 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.03 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right ) \, dx=x^4\,\left (\frac {A\,b\,e^2}{4}+\frac {B\,c\,d^2}{4}+\frac {A\,c\,d\,e}{2}+\frac {B\,b\,d\,e}{2}\right )+x^3\,\left (\frac {A\,c\,d^2}{3}+\frac {B\,b\,d^2}{3}+\frac {2\,A\,b\,d\,e}{3}\right )+x^5\,\left (\frac {A\,c\,e^2}{5}+\frac {B\,b\,e^2}{5}+\frac {2\,B\,c\,d\,e}{5}\right )+\frac {A\,b\,d^2\,x^2}{2}+\frac {B\,c\,e^2\,x^6}{6} \] Input:
int((b*x + c*x^2)*(A + B*x)*(d + e*x)^2,x)
Output:
x^4*((A*b*e^2)/4 + (B*c*d^2)/4 + (A*c*d*e)/2 + (B*b*d*e)/2) + x^3*((A*c*d^ 2)/3 + (B*b*d^2)/3 + (2*A*b*d*e)/3) + x^5*((A*c*e^2)/5 + (B*b*e^2)/5 + (2* B*c*d*e)/5) + (A*b*d^2*x^2)/2 + (B*c*e^2*x^6)/6
Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.17 \[ \int (A+B x) (d+e x)^2 \left (b x+c x^2\right ) \, dx=\frac {x^{2} \left (10 b c \,e^{2} x^{4}+12 a c \,e^{2} x^{3}+12 b^{2} e^{2} x^{3}+24 b c d e \,x^{3}+15 a b \,e^{2} x^{2}+30 a c d e \,x^{2}+30 b^{2} d e \,x^{2}+15 b c \,d^{2} x^{2}+40 a b d e x +20 a c \,d^{2} x +20 b^{2} d^{2} x +30 a b \,d^{2}\right )}{60} \] Input:
int((B*x+A)*(e*x+d)^2*(c*x^2+b*x),x)
Output:
(x**2*(30*a*b*d**2 + 40*a*b*d*e*x + 15*a*b*e**2*x**2 + 20*a*c*d**2*x + 30* a*c*d*e*x**2 + 12*a*c*e**2*x**3 + 20*b**2*d**2*x + 30*b**2*d*e*x**2 + 12*b **2*e**2*x**3 + 15*b*c*d**2*x**2 + 24*b*c*d*e*x**3 + 10*b*c*e**2*x**4))/60