Integrand size = 18, antiderivative size = 96 \[ \int (A+B x) \left (a+b x+c x^2\right )^2 \, dx=a^2 A x+\frac {1}{2} a (2 A b+a B) x^2+\frac {1}{3} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^3+\frac {1}{4} \left (b^2 B+2 A b c+2 a B c\right ) x^4+\frac {1}{5} c (2 b B+A c) x^5+\frac {1}{6} B c^2 x^6 \] Output:
a^2*A*x+1/2*a*(2*A*b+B*a)*x^2+1/3*(2*a*b*B+A*(2*a*c+b^2))*x^3+1/4*(2*A*b*c +2*B*a*c+B*b^2)*x^4+1/5*c*(A*c+2*B*b)*x^5+1/6*B*c^2*x^6
Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00 \[ \int (A+B x) \left (a+b x+c x^2\right )^2 \, dx=a^2 A x+\frac {1}{2} a (2 A b+a B) x^2+\frac {1}{3} \left (A b^2+2 a b B+2 a A c\right ) x^3+\frac {1}{4} \left (b^2 B+2 A b c+2 a B c\right ) x^4+\frac {1}{5} c (2 b B+A c) x^5+\frac {1}{6} B c^2 x^6 \] Input:
Integrate[(A + B*x)*(a + b*x + c*x^2)^2,x]
Output:
a^2*A*x + (a*(2*A*b + a*B)*x^2)/2 + ((A*b^2 + 2*a*b*B + 2*a*A*c)*x^3)/3 + ((b^2*B + 2*A*b*c + 2*a*B*c)*x^4)/4 + (c*(2*b*B + A*c)*x^5)/5 + (B*c^2*x^6 )/6
Time = 0.28 (sec) , antiderivative size = 96, 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 = {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.
\(\displaystyle \int (A+B x) \left (a+b x+c x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (a^2 A+x^3 \left (2 a B c+2 A b c+b^2 B\right )+x^2 \left (A \left (2 a c+b^2\right )+2 a b B\right )+a x (a B+2 A b)+c x^4 (A c+2 b B)+B c^2 x^5\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 A x+\frac {1}{4} x^4 \left (2 a B c+2 A b c+b^2 B\right )+\frac {1}{3} x^3 \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {1}{2} a x^2 (a B+2 A b)+\frac {1}{5} c x^5 (A c+2 b B)+\frac {1}{6} B c^2 x^6\) |
Input:
Int[(A + B*x)*(a + b*x + c*x^2)^2,x]
Output:
a^2*A*x + (a*(2*A*b + a*B)*x^2)/2 + ((2*a*b*B + A*(b^2 + 2*a*c))*x^3)/3 + ((b^2*B + 2*A*b*c + 2*a*B*c)*x^4)/4 + (c*(2*b*B + A*c)*x^5)/5 + (B*c^2*x^6 )/6
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 = 1.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94
method | result | size |
norman | \(\frac {B \,c^{2} x^{6}}{6}+\left (\frac {1}{5} A \,c^{2}+\frac {2}{5} B b c \right ) x^{5}+\left (\frac {1}{2} A b c +\frac {1}{2} a B c +\frac {1}{4} B \,b^{2}\right ) x^{4}+\left (\frac {2}{3} A a c +\frac {1}{3} b^{2} A +\frac {2}{3} a b B \right ) x^{3}+\left (a b A +\frac {1}{2} a^{2} B \right ) x^{2}+a^{2} A x\) | \(90\) |
default | \(\frac {B \,c^{2} x^{6}}{6}+\frac {\left (A \,c^{2}+2 B b c \right ) x^{5}}{5}+\frac {\left (2 A b c +B \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (2 a b B +A \left (2 a c +b^{2}\right )\right ) x^{3}}{3}+\frac {\left (2 a b A +a^{2} B \right ) x^{2}}{2}+a^{2} A x\) | \(91\) |
gosper | \(\frac {1}{6} B \,c^{2} x^{6}+\frac {1}{5} A \,c^{2} x^{5}+\frac {2}{5} x^{5} B b c +\frac {1}{2} x^{4} A b c +\frac {1}{2} B a c \,x^{4}+\frac {1}{4} x^{4} B \,b^{2}+\frac {2}{3} a A c \,x^{3}+\frac {1}{3} A \,x^{3} b^{2}+\frac {2}{3} a b B \,x^{3}+a b A \,x^{2}+\frac {1}{2} B \,a^{2} x^{2}+a^{2} A x\) | \(100\) |
risch | \(\frac {1}{6} B \,c^{2} x^{6}+\frac {1}{5} A \,c^{2} x^{5}+\frac {2}{5} x^{5} B b c +\frac {1}{2} x^{4} A b c +\frac {1}{2} B a c \,x^{4}+\frac {1}{4} x^{4} B \,b^{2}+\frac {2}{3} a A c \,x^{3}+\frac {1}{3} A \,x^{3} b^{2}+\frac {2}{3} a b B \,x^{3}+a b A \,x^{2}+\frac {1}{2} B \,a^{2} x^{2}+a^{2} A x\) | \(100\) |
parallelrisch | \(\frac {1}{6} B \,c^{2} x^{6}+\frac {1}{5} A \,c^{2} x^{5}+\frac {2}{5} x^{5} B b c +\frac {1}{2} x^{4} A b c +\frac {1}{2} B a c \,x^{4}+\frac {1}{4} x^{4} B \,b^{2}+\frac {2}{3} a A c \,x^{3}+\frac {1}{3} A \,x^{3} b^{2}+\frac {2}{3} a b B \,x^{3}+a b A \,x^{2}+\frac {1}{2} B \,a^{2} x^{2}+a^{2} A x\) | \(100\) |
orering | \(\frac {x \left (10 B \,c^{2} x^{5}+12 x^{4} A \,c^{2}+24 x^{4} B b c +30 x^{3} A b c +30 B a c \,x^{3}+15 x^{3} B \,b^{2}+40 A a c \,x^{2}+20 x^{2} b^{2} A +40 B a \,x^{2} b +60 a b A x +30 a^{2} B x +60 a^{2} A \right )}{60}\) | \(100\) |
Input:
int((B*x+A)*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/6*B*c^2*x^6+(1/5*A*c^2+2/5*B*b*c)*x^5+(1/2*A*b*c+1/2*a*B*c+1/4*B*b^2)*x^ 4+(2/3*A*a*c+1/3*b^2*A+2/3*a*b*B)*x^3+(a*b*A+1/2*a^2*B)*x^2+a^2*A*x
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{6} \, B c^{2} x^{6} + \frac {1}{5} \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{4} + A a^{2} x + \frac {1}{3} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} x^{2} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
1/6*B*c^2*x^6 + 1/5*(2*B*b*c + A*c^2)*x^5 + 1/4*(B*b^2 + 2*(B*a + A*b)*c)* x^4 + A*a^2*x + 1/3*(2*B*a*b + A*b^2 + 2*A*a*c)*x^3 + 1/2*(B*a^2 + 2*A*a*b )*x^2
Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.04 \[ \int (A+B x) \left (a+b x+c x^2\right )^2 \, dx=A a^{2} x + \frac {B c^{2} x^{6}}{6} + x^{5} \left (\frac {A c^{2}}{5} + \frac {2 B b c}{5}\right ) + x^{4} \left (\frac {A b c}{2} + \frac {B a c}{2} + \frac {B b^{2}}{4}\right ) + x^{3} \cdot \left (\frac {2 A a c}{3} + \frac {A b^{2}}{3} + \frac {2 B a b}{3}\right ) + x^{2} \left (A a b + \frac {B a^{2}}{2}\right ) \] Input:
integrate((B*x+A)*(c*x**2+b*x+a)**2,x)
Output:
A*a**2*x + B*c**2*x**6/6 + x**5*(A*c**2/5 + 2*B*b*c/5) + x**4*(A*b*c/2 + B *a*c/2 + B*b**2/4) + x**3*(2*A*a*c/3 + A*b**2/3 + 2*B*a*b/3) + x**2*(A*a*b + B*a**2/2)
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{6} \, B c^{2} x^{6} + \frac {1}{5} \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{4} + A a^{2} x + \frac {1}{3} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} x^{2} \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^2,x, algorithm="maxima")
Output:
1/6*B*c^2*x^6 + 1/5*(2*B*b*c + A*c^2)*x^5 + 1/4*(B*b^2 + 2*(B*a + A*b)*c)* x^4 + A*a^2*x + 1/3*(2*B*a*b + A*b^2 + 2*A*a*c)*x^3 + 1/2*(B*a^2 + 2*A*a*b )*x^2
Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03 \[ \int (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{6} \, B c^{2} x^{6} + \frac {2}{5} \, B b c x^{5} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{4} \, B b^{2} x^{4} + \frac {1}{2} \, B a c x^{4} + \frac {1}{2} \, A b c x^{4} + \frac {2}{3} \, B a b x^{3} + \frac {1}{3} \, A b^{2} x^{3} + \frac {2}{3} \, A a c x^{3} + \frac {1}{2} \, B a^{2} x^{2} + A a b x^{2} + A a^{2} x \] Input:
integrate((B*x+A)*(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
1/6*B*c^2*x^6 + 2/5*B*b*c*x^5 + 1/5*A*c^2*x^5 + 1/4*B*b^2*x^4 + 1/2*B*a*c* x^4 + 1/2*A*b*c*x^4 + 2/3*B*a*b*x^3 + 1/3*A*b^2*x^3 + 2/3*A*a*c*x^3 + 1/2* B*a^2*x^2 + A*a*b*x^2 + A*a^2*x
Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.93 \[ \int (A+B x) \left (a+b x+c x^2\right )^2 \, dx=x^2\,\left (\frac {B\,a^2}{2}+A\,b\,a\right )+x^5\,\left (\frac {A\,c^2}{5}+\frac {2\,B\,b\,c}{5}\right )+x^3\,\left (\frac {A\,b^2}{3}+\frac {2\,B\,a\,b}{3}+\frac {2\,A\,a\,c}{3}\right )+x^4\,\left (\frac {B\,b^2}{4}+\frac {A\,c\,b}{2}+\frac {B\,a\,c}{2}\right )+\frac {B\,c^2\,x^6}{6}+A\,a^2\,x \] Input:
int((A + B*x)*(a + b*x + c*x^2)^2,x)
Output:
x^2*((B*a^2)/2 + A*a*b) + x^5*((A*c^2)/5 + (2*B*b*c)/5) + x^3*((A*b^2)/3 + (2*A*a*c)/3 + (2*B*a*b)/3) + x^4*((B*b^2)/4 + (A*b*c)/2 + (B*a*c)/2) + (B *c^2*x^6)/6 + A*a^2*x
Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.80 \[ \int (A+B x) \left (a+b x+c x^2\right )^2 \, dx=\frac {x \left (10 b \,c^{2} x^{5}+12 a \,c^{2} x^{4}+24 b^{2} c \,x^{4}+60 a b c \,x^{3}+15 b^{3} x^{3}+40 a^{2} c \,x^{2}+60 a \,b^{2} x^{2}+90 a^{2} b x +60 a^{3}\right )}{60} \] Input:
int((B*x+A)*(c*x^2+b*x+a)^2,x)
Output:
(x*(60*a**3 + 90*a**2*b*x + 40*a**2*c*x**2 + 60*a*b**2*x**2 + 60*a*b*c*x** 3 + 12*a*c**2*x**4 + 15*b**3*x**3 + 24*b**2*c*x**4 + 10*b*c**2*x**5))/60