Integrand size = 18, antiderivative size = 75 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=\frac {(A b-a B) (b d-a e) (a+b x)^3}{3 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^4}{4 b^3}+\frac {B e (a+b x)^5}{5 b^3} \] Output:
1/3*(A*b-B*a)*(-a*e+b*d)*(b*x+a)^3/b^3+1/4*(A*b*e-2*B*a*e+B*b*d)*(b*x+a)^4 /b^3+1/5*B*e*(b*x+a)^5/b^3
Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.28 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=a^2 A d x+\frac {1}{2} a (2 A b d+a B d+a A e) x^2+\frac {1}{3} \left (A b^2 d+2 a b B d+2 a A b e+a^2 B e\right ) x^3+\frac {1}{4} b (b B d+A b e+2 a B e) x^4+\frac {1}{5} b^2 B e x^5 \] Input:
Integrate[(a + b*x)^2*(A + B*x)*(d + e*x),x]
Output:
a^2*A*d*x + (a*(2*A*b*d + a*B*d + a*A*e)*x^2)/2 + ((A*b^2*d + 2*a*b*B*d + 2*a*A*b*e + a^2*B*e)*x^3)/3 + (b*(b*B*d + A*b*e + 2*a*B*e)*x^4)/4 + (b^2*B *e*x^5)/5
Time = 0.28 (sec) , antiderivative size = 75, 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.
\(\displaystyle \int (a+b x)^2 (A+B x) (d+e x) \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {(a+b x)^3 (-2 a B e+A b e+b B d)}{b^2}+\frac {(a+b x)^2 (A b-a B) (b d-a e)}{b^2}+\frac {B e (a+b x)^4}{b^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(a+b x)^4 (-2 a B e+A b e+b B d)}{4 b^3}+\frac {(a+b x)^3 (A b-a B) (b d-a e)}{3 b^3}+\frac {B e (a+b x)^5}{5 b^3}\) |
Input:
Int[(a + b*x)^2*(A + B*x)*(d + e*x),x]
Output:
((A*b - a*B)*(b*d - a*e)*(a + b*x)^3)/(3*b^3) + ((b*B*d + A*b*e - 2*a*B*e) *(a + b*x)^4)/(4*b^3) + (B*e*(a + b*x)^5)/(5*b^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 = 0.15 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.32
method | result | size |
norman | \(\frac {b^{2} B e \,x^{5}}{5}+\left (\frac {1}{4} A \,b^{2} e +\frac {1}{2} B a b e +\frac {1}{4} b^{2} B d \right ) x^{4}+\left (\frac {2}{3} A a b e +\frac {1}{3} A \,b^{2} d +\frac {1}{3} B \,a^{2} e +\frac {2}{3} B a b d \right ) x^{3}+\left (\frac {1}{2} a^{2} A e +A a b d +\frac {1}{2} B \,a^{2} d \right ) x^{2}+a^{2} A d x\) | \(99\) |
default | \(\frac {b^{2} B e \,x^{5}}{5}+\frac {\left (\left (b^{2} A +2 a b B \right ) e +b^{2} B d \right ) x^{4}}{4}+\frac {\left (\left (2 a b A +a^{2} B \right ) e +\left (b^{2} A +2 a b B \right ) d \right ) x^{3}}{3}+\frac {\left (a^{2} A e +\left (2 a b A +a^{2} B \right ) d \right ) x^{2}}{2}+a^{2} A d x\) | \(101\) |
orering | \(\frac {x \left (12 b^{2} B e \,x^{4}+15 A \,b^{2} e \,x^{3}+30 B a b e \,x^{3}+15 B \,b^{2} d \,x^{3}+40 A a b e \,x^{2}+20 A \,b^{2} d \,x^{2}+20 B \,a^{2} e \,x^{2}+40 B a b d \,x^{2}+30 A \,a^{2} e x +60 A a b d x +30 B \,a^{2} d x +60 a^{2} A d \right )}{60}\) | \(112\) |
gosper | \(\frac {1}{5} b^{2} B e \,x^{5}+\frac {1}{4} x^{4} A \,b^{2} e +\frac {1}{2} x^{4} B a b e +\frac {1}{4} x^{4} b^{2} B d +\frac {2}{3} x^{3} A a b e +\frac {1}{3} x^{3} A \,b^{2} d +\frac {1}{3} x^{3} B \,a^{2} e +\frac {2}{3} x^{3} B a b d +\frac {1}{2} x^{2} a^{2} A e +x^{2} A a b d +\frac {1}{2} x^{2} B \,a^{2} d +a^{2} A d x\) | \(114\) |
risch | \(\frac {1}{5} b^{2} B e \,x^{5}+\frac {1}{4} x^{4} A \,b^{2} e +\frac {1}{2} x^{4} B a b e +\frac {1}{4} x^{4} b^{2} B d +\frac {2}{3} x^{3} A a b e +\frac {1}{3} x^{3} A \,b^{2} d +\frac {1}{3} x^{3} B \,a^{2} e +\frac {2}{3} x^{3} B a b d +\frac {1}{2} x^{2} a^{2} A e +x^{2} A a b d +\frac {1}{2} x^{2} B \,a^{2} d +a^{2} A d x\) | \(114\) |
parallelrisch | \(\frac {1}{5} b^{2} B e \,x^{5}+\frac {1}{4} x^{4} A \,b^{2} e +\frac {1}{2} x^{4} B a b e +\frac {1}{4} x^{4} b^{2} B d +\frac {2}{3} x^{3} A a b e +\frac {1}{3} x^{3} A \,b^{2} d +\frac {1}{3} x^{3} B \,a^{2} e +\frac {2}{3} x^{3} B a b d +\frac {1}{2} x^{2} a^{2} A e +x^{2} A a b d +\frac {1}{2} x^{2} B \,a^{2} d +a^{2} A d x\) | \(114\) |
Input:
int((b*x+a)^2*(B*x+A)*(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/5*b^2*B*e*x^5+(1/4*A*b^2*e+1/2*B*a*b*e+1/4*b^2*B*d)*x^4+(2/3*A*a*b*e+1/3 *A*b^2*d+1/3*B*a^2*e+2/3*B*a*b*d)*x^3+(1/2*a^2*A*e+A*a*b*d+1/2*B*a^2*d)*x^ 2+a^2*A*d*x
Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.33 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=\frac {1}{5} \, B b^{2} e x^{5} + A a^{2} d x + \frac {1}{4} \, {\left (B b^{2} d + {\left (2 \, B a b + A b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left ({\left (2 \, B a b + A b^{2}\right )} d + {\left (B a^{2} + 2 \, A a b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d\right )} x^{2} \] Input:
integrate((b*x+a)^2*(B*x+A)*(e*x+d),x, algorithm="fricas")
Output:
1/5*B*b^2*e*x^5 + A*a^2*d*x + 1/4*(B*b^2*d + (2*B*a*b + A*b^2)*e)*x^4 + 1/ 3*((2*B*a*b + A*b^2)*d + (B*a^2 + 2*A*a*b)*e)*x^3 + 1/2*(A*a^2*e + (B*a^2 + 2*A*a*b)*d)*x^2
Time = 0.02 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.55 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=A a^{2} d x + \frac {B b^{2} e x^{5}}{5} + x^{4} \left (\frac {A b^{2} e}{4} + \frac {B a b e}{2} + \frac {B b^{2} d}{4}\right ) + x^{3} \cdot \left (\frac {2 A a b e}{3} + \frac {A b^{2} d}{3} + \frac {B a^{2} e}{3} + \frac {2 B a b d}{3}\right ) + x^{2} \left (\frac {A a^{2} e}{2} + A a b d + \frac {B a^{2} d}{2}\right ) \] Input:
integrate((b*x+a)**2*(B*x+A)*(e*x+d),x)
Output:
A*a**2*d*x + B*b**2*e*x**5/5 + x**4*(A*b**2*e/4 + B*a*b*e/2 + B*b**2*d/4) + x**3*(2*A*a*b*e/3 + A*b**2*d/3 + B*a**2*e/3 + 2*B*a*b*d/3) + x**2*(A*a** 2*e/2 + A*a*b*d + B*a**2*d/2)
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.33 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=\frac {1}{5} \, B b^{2} e x^{5} + A a^{2} d x + \frac {1}{4} \, {\left (B b^{2} d + {\left (2 \, B a b + A b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left ({\left (2 \, B a b + A b^{2}\right )} d + {\left (B a^{2} + 2 \, A a b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d\right )} x^{2} \] Input:
integrate((b*x+a)^2*(B*x+A)*(e*x+d),x, algorithm="maxima")
Output:
1/5*B*b^2*e*x^5 + A*a^2*d*x + 1/4*(B*b^2*d + (2*B*a*b + A*b^2)*e)*x^4 + 1/ 3*((2*B*a*b + A*b^2)*d + (B*a^2 + 2*A*a*b)*e)*x^3 + 1/2*(A*a^2*e + (B*a^2 + 2*A*a*b)*d)*x^2
Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.51 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=\frac {1}{5} \, B b^{2} e x^{5} + \frac {1}{4} \, B b^{2} d x^{4} + \frac {1}{2} \, B a b e x^{4} + \frac {1}{4} \, A b^{2} e x^{4} + \frac {2}{3} \, B a b d x^{3} + \frac {1}{3} \, A b^{2} d x^{3} + \frac {1}{3} \, B a^{2} e x^{3} + \frac {2}{3} \, A a b e x^{3} + \frac {1}{2} \, B a^{2} d x^{2} + A a b d x^{2} + \frac {1}{2} \, A a^{2} e x^{2} + A a^{2} d x \] Input:
integrate((b*x+a)^2*(B*x+A)*(e*x+d),x, algorithm="giac")
Output:
1/5*B*b^2*e*x^5 + 1/4*B*b^2*d*x^4 + 1/2*B*a*b*e*x^4 + 1/4*A*b^2*e*x^4 + 2/ 3*B*a*b*d*x^3 + 1/3*A*b^2*d*x^3 + 1/3*B*a^2*e*x^3 + 2/3*A*a*b*e*x^3 + 1/2* B*a^2*d*x^2 + A*a*b*d*x^2 + 1/2*A*a^2*e*x^2 + A*a^2*d*x
Time = 0.90 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.31 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=x^3\,\left (\frac {A\,b^2\,d}{3}+\frac {B\,a^2\,e}{3}+\frac {2\,A\,a\,b\,e}{3}+\frac {2\,B\,a\,b\,d}{3}\right )+x^2\,\left (\frac {A\,a^2\,e}{2}+\frac {B\,a^2\,d}{2}+A\,a\,b\,d\right )+x^4\,\left (\frac {A\,b^2\,e}{4}+\frac {B\,b^2\,d}{4}+\frac {B\,a\,b\,e}{2}\right )+A\,a^2\,d\,x+\frac {B\,b^2\,e\,x^5}{5} \] Input:
int((A + B*x)*(a + b*x)^2*(d + e*x),x)
Output:
x^3*((A*b^2*d)/3 + (B*a^2*e)/3 + (2*A*a*b*e)/3 + (2*B*a*b*d)/3) + x^2*((A* a^2*e)/2 + (B*a^2*d)/2 + A*a*b*d) + x^4*((A*b^2*e)/4 + (B*b^2*d)/4 + (B*a* b*e)/2) + A*a^2*d*x + (B*b^2*e*x^5)/5
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int (a+b x)^2 (A+B x) (d+e x) \, dx=\frac {x \left (4 b^{3} e \,x^{4}+15 a \,b^{2} e \,x^{3}+5 b^{3} d \,x^{3}+20 a^{2} b e \,x^{2}+20 a \,b^{2} d \,x^{2}+10 a^{3} e x +30 a^{2} b d x +20 a^{3} d \right )}{20} \] Input:
int((b*x+a)^2*(B*x+A)*(e*x+d),x)
Output:
(x*(20*a**3*d + 10*a**3*e*x + 30*a**2*b*d*x + 20*a**2*b*e*x**2 + 20*a*b**2 *d*x**2 + 15*a*b**2*e*x**3 + 5*b**3*d*x**3 + 4*b**3*e*x**4))/20