Integrand size = 18, antiderivative size = 77 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=\frac {(b d-a e) (B d-A e) (d+e x)^3}{3 e^3}-\frac {(2 b B d-A b e-a B e) (d+e x)^4}{4 e^3}+\frac {b B (d+e x)^5}{5 e^3} \] Output:
1/3*(-a*e+b*d)*(-A*e+B*d)*(e*x+d)^3/e^3-1/4*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d) ^4/e^3+1/5*b*B*(e*x+d)^5/e^3
Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.25 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=a A d^2 x+\frac {1}{2} d (A b d+a B d+2 a A e) x^2+\frac {1}{3} \left (b B d^2+2 A b d e+2 a B d e+a A e^2\right ) x^3+\frac {1}{4} e (2 b B d+A b e+a B e) x^4+\frac {1}{5} b B e^2 x^5 \] Input:
Integrate[(a + b*x)*(A + B*x)*(d + e*x)^2,x]
Output:
a*A*d^2*x + (d*(A*b*d + a*B*d + 2*a*A*e)*x^2)/2 + ((b*B*d^2 + 2*A*b*d*e + 2*a*B*d*e + a*A*e^2)*x^3)/3 + (e*(2*b*B*d + A*b*e + a*B*e)*x^4)/4 + (b*B*e ^2*x^5)/5
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {86, 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.
\(\displaystyle \int (a+b x) (A+B x) (d+e x)^2 \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {(d+e x)^3 (a B e+A b e-2 b B d)}{e^2}+\frac {(d+e x)^2 (a e-b d) (A e-B d)}{e^2}+\frac {b B (d+e x)^4}{e^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(d+e x)^4 (-a B e-A b e+2 b B d)}{4 e^3}+\frac {(d+e x)^3 (b d-a e) (B d-A e)}{3 e^3}+\frac {b B (d+e x)^5}{5 e^3}\) |
Input:
Int[(a + b*x)*(A + B*x)*(d + e*x)^2,x]
Output:
((b*d - a*e)*(B*d - A*e)*(d + e*x)^3)/(3*e^3) - ((2*b*B*d - A*b*e - a*B*e) *(d + e*x)^4)/(4*e^3) + (b*B*(d + e*x)^5)/(5*e^3)
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.22
method | result | size |
default | \(\frac {b B \,e^{2} x^{5}}{5}+\frac {\left (\left (A b +B a \right ) e^{2}+2 b B d e \right ) x^{4}}{4}+\frac {\left (A a \,e^{2}+2 \left (A b +B a \right ) d e +b B \,d^{2}\right ) x^{3}}{3}+\frac {\left (2 A a d e +\left (A b +B a \right ) d^{2}\right ) x^{2}}{2}+A a \,d^{2} x\) | \(94\) |
norman | \(\frac {b B \,e^{2} x^{5}}{5}+\left (\frac {1}{4} A b \,e^{2}+\frac {1}{4} B a \,e^{2}+\frac {1}{2} b B d e \right ) x^{4}+\left (\frac {1}{3} A a \,e^{2}+\frac {2}{3} A b d e +\frac {2}{3} B a d e +\frac {1}{3} b B \,d^{2}\right ) x^{3}+\left (A a d e +\frac {1}{2} A b \,d^{2}+\frac {1}{2} B a \,d^{2}\right ) x^{2}+A a \,d^{2} x\) | \(99\) |
orering | \(\frac {x \left (12 b B \,e^{2} x^{4}+15 A b \,e^{2} x^{3}+15 B a \,e^{2} x^{3}+30 B b d e \,x^{3}+20 A a \,e^{2} x^{2}+40 A b d e \,x^{2}+40 B a d e \,x^{2}+20 B b \,d^{2} x^{2}+60 A a d e x +30 A b \,d^{2} x +30 B a \,d^{2} x +60 A a \,d^{2}\right )}{60}\) | \(112\) |
gosper | \(\frac {1}{5} b B \,e^{2} x^{5}+\frac {1}{4} x^{4} A b \,e^{2}+\frac {1}{4} x^{4} B a \,e^{2}+\frac {1}{2} x^{4} b B d e +\frac {1}{3} x^{3} A a \,e^{2}+\frac {2}{3} x^{3} A b d e +\frac {2}{3} x^{3} B a d e +\frac {1}{3} x^{3} b B \,d^{2}+x^{2} A a d e +\frac {1}{2} x^{2} A b \,d^{2}+\frac {1}{2} x^{2} B a \,d^{2}+A a \,d^{2} x\) | \(114\) |
risch | \(\frac {1}{5} b B \,e^{2} x^{5}+\frac {1}{4} x^{4} A b \,e^{2}+\frac {1}{4} x^{4} B a \,e^{2}+\frac {1}{2} x^{4} b B d e +\frac {1}{3} x^{3} A a \,e^{2}+\frac {2}{3} x^{3} A b d e +\frac {2}{3} x^{3} B a d e +\frac {1}{3} x^{3} b B \,d^{2}+x^{2} A a d e +\frac {1}{2} x^{2} A b \,d^{2}+\frac {1}{2} x^{2} B a \,d^{2}+A a \,d^{2} x\) | \(114\) |
parallelrisch | \(\frac {1}{5} b B \,e^{2} x^{5}+\frac {1}{4} x^{4} A b \,e^{2}+\frac {1}{4} x^{4} B a \,e^{2}+\frac {1}{2} x^{4} b B d e +\frac {1}{3} x^{3} A a \,e^{2}+\frac {2}{3} x^{3} A b d e +\frac {2}{3} x^{3} B a d e +\frac {1}{3} x^{3} b B \,d^{2}+x^{2} A a d e +\frac {1}{2} x^{2} A b \,d^{2}+\frac {1}{2} x^{2} B a \,d^{2}+A a \,d^{2} x\) | \(114\) |
Input:
int((b*x+a)*(B*x+A)*(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
1/5*b*B*e^2*x^5+1/4*((A*b+B*a)*e^2+2*b*B*d*e)*x^4+1/3*(A*a*e^2+2*(A*b+B*a) *d*e+b*B*d^2)*x^3+1/2*(2*A*a*d*e+(A*b+B*a)*d^2)*x^2+A*a*d^2*x
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.21 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=\frac {1}{5} \, B b e^{2} x^{5} + A a d^{2} x + \frac {1}{4} \, {\left (2 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b d^{2} + A a e^{2} + 2 \, {\left (B a + A b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a d e + {\left (B a + A b\right )} d^{2}\right )} x^{2} \] Input:
integrate((b*x+a)*(B*x+A)*(e*x+d)^2,x, algorithm="fricas")
Output:
1/5*B*b*e^2*x^5 + A*a*d^2*x + 1/4*(2*B*b*d*e + (B*a + A*b)*e^2)*x^4 + 1/3* (B*b*d^2 + A*a*e^2 + 2*(B*a + A*b)*d*e)*x^3 + 1/2*(2*A*a*d*e + (B*a + A*b) *d^2)*x^2
Time = 0.02 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.51 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=A a d^{2} x + \frac {B b e^{2} x^{5}}{5} + x^{4} \left (\frac {A b e^{2}}{4} + \frac {B a e^{2}}{4} + \frac {B b d e}{2}\right ) + x^{3} \left (\frac {A a e^{2}}{3} + \frac {2 A b d e}{3} + \frac {2 B a d e}{3} + \frac {B b d^{2}}{3}\right ) + x^{2} \left (A a d e + \frac {A b d^{2}}{2} + \frac {B a d^{2}}{2}\right ) \] Input:
integrate((b*x+a)*(B*x+A)*(e*x+d)**2,x)
Output:
A*a*d**2*x + B*b*e**2*x**5/5 + x**4*(A*b*e**2/4 + B*a*e**2/4 + B*b*d*e/2) + x**3*(A*a*e**2/3 + 2*A*b*d*e/3 + 2*B*a*d*e/3 + B*b*d**2/3) + x**2*(A*a*d *e + A*b*d**2/2 + B*a*d**2/2)
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.21 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=\frac {1}{5} \, B b e^{2} x^{5} + A a d^{2} x + \frac {1}{4} \, {\left (2 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b d^{2} + A a e^{2} + 2 \, {\left (B a + A b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a d e + {\left (B a + A b\right )} d^{2}\right )} x^{2} \] Input:
integrate((b*x+a)*(B*x+A)*(e*x+d)^2,x, algorithm="maxima")
Output:
1/5*B*b*e^2*x^5 + A*a*d^2*x + 1/4*(2*B*b*d*e + (B*a + A*b)*e^2)*x^4 + 1/3* (B*b*d^2 + A*a*e^2 + 2*(B*a + A*b)*d*e)*x^3 + 1/2*(2*A*a*d*e + (B*a + A*b) *d^2)*x^2
Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.47 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=\frac {1}{5} \, B b e^{2} x^{5} + \frac {1}{2} \, B b d e x^{4} + \frac {1}{4} \, B a e^{2} x^{4} + \frac {1}{4} \, A b e^{2} x^{4} + \frac {1}{3} \, B b d^{2} x^{3} + \frac {2}{3} \, B a d e x^{3} + \frac {2}{3} \, A b d e x^{3} + \frac {1}{3} \, A a e^{2} x^{3} + \frac {1}{2} \, B a d^{2} x^{2} + \frac {1}{2} \, A b d^{2} x^{2} + A a d e x^{2} + A a d^{2} x \] Input:
integrate((b*x+a)*(B*x+A)*(e*x+d)^2,x, algorithm="giac")
Output:
1/5*B*b*e^2*x^5 + 1/2*B*b*d*e*x^4 + 1/4*B*a*e^2*x^4 + 1/4*A*b*e^2*x^4 + 1/ 3*B*b*d^2*x^3 + 2/3*B*a*d*e*x^3 + 2/3*A*b*d*e*x^3 + 1/3*A*a*e^2*x^3 + 1/2* B*a*d^2*x^2 + 1/2*A*b*d^2*x^2 + A*a*d*e*x^2 + A*a*d^2*x
Time = 0.87 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.27 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=x^3\,\left (\frac {A\,a\,e^2}{3}+\frac {B\,b\,d^2}{3}+\frac {2\,A\,b\,d\,e}{3}+\frac {2\,B\,a\,d\,e}{3}\right )+x^2\,\left (\frac {A\,b\,d^2}{2}+\frac {B\,a\,d^2}{2}+A\,a\,d\,e\right )+x^4\,\left (\frac {A\,b\,e^2}{4}+\frac {B\,a\,e^2}{4}+\frac {B\,b\,d\,e}{2}\right )+A\,a\,d^2\,x+\frac {B\,b\,e^2\,x^5}{5} \] Input:
int((A + B*x)*(a + b*x)*(d + e*x)^2,x)
Output:
x^3*((A*a*e^2)/3 + (B*b*d^2)/3 + (2*A*b*d*e)/3 + (2*B*a*d*e)/3) + x^2*((A* b*d^2)/2 + (B*a*d^2)/2 + A*a*d*e) + x^4*((A*b*e^2)/4 + (B*a*e^2)/4 + (B*b* d*e)/2) + A*a*d^2*x + (B*b*e^2*x^5)/5
Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.17 \[ \int (a+b x) (A+B x) (d+e x)^2 \, dx=\frac {x \left (6 b^{2} e^{2} x^{4}+15 a b \,e^{2} x^{3}+15 b^{2} d e \,x^{3}+10 a^{2} e^{2} x^{2}+40 a b d e \,x^{2}+10 b^{2} d^{2} x^{2}+30 a^{2} d e x +30 a b \,d^{2} x +30 a^{2} d^{2}\right )}{30} \] Input:
int((b*x+a)*(B*x+A)*(e*x+d)^2,x)
Output:
(x*(30*a**2*d**2 + 30*a**2*d*e*x + 10*a**2*e**2*x**2 + 30*a*b*d**2*x + 40* a*b*d*e*x**2 + 15*a*b*e**2*x**3 + 10*b**2*d**2*x**2 + 15*b**2*d*e*x**3 + 6 *b**2*e**2*x**4))/30