Integrand size = 23, antiderivative size = 180 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=a^2 A d x+\frac {1}{2} a (2 A b d+a B d+a A e) x^2+\frac {1}{3} \left (a B (2 b d+a e)+A \left (b^2 d+2 a c d+2 a b e\right )\right ) x^3+\frac {1}{4} \left (b^2 (B d+A e)+2 a c (B d+A e)+2 b (A c d+a B e)\right ) x^4+\frac {1}{5} \left (b^2 B e+2 b c (B d+A e)+c (A c d+2 a B e)\right ) x^5+\frac {1}{6} c (B c d+2 b B e+A c e) x^6+\frac {1}{7} B c^2 e x^7 \]
a^2*A*d*x+1/2*a*(A*a*e+2*A*b*d+B*a*d)*x^2+1/3*(a*B*(a*e+2*b*d)+A*(2*a*b*e+ 2*a*c*d+b^2*d))*x^3+1/4*(b^2*(A*e+B*d)+2*a*c*(A*e+B*d)+2*b*(A*c*d+B*a*e))* x^4+1/5*(b^2*B*e+2*b*c*(A*e+B*d)+c*(A*c*d+2*B*a*e))*x^5+1/6*c*(A*c*e+2*B*b *e+B*c*d)*x^6+1/7*B*c^2*e*x^7
Time = 0.05 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=a^2 A d x+\frac {1}{2} a (2 A b d+a B d+a A e) x^2+\frac {1}{3} \left (a B (2 b d+a e)+A \left (b^2 d+2 a c d+2 a b e\right )\right ) x^3+\frac {1}{4} \left (b^2 (B d+A e)+2 a c (B d+A e)+2 b (A c d+a B e)\right ) x^4+\frac {1}{5} \left (b^2 B e+2 b c (B d+A e)+c (A c d+2 a B e)\right ) x^5+\frac {1}{6} c (B c d+2 b B e+A c e) x^6+\frac {1}{7} B c^2 e x^7 \]
a^2*A*d*x + (a*(2*A*b*d + a*B*d + a*A*e)*x^2)/2 + ((a*B*(2*b*d + a*e) + A* (b^2*d + 2*a*c*d + 2*a*b*e))*x^3)/3 + ((b^2*(B*d + A*e) + 2*a*c*(B*d + A*e ) + 2*b*(A*c*d + a*B*e))*x^4)/4 + ((b^2*B*e + 2*b*c*(B*d + A*e) + c*(A*c*d + 2*a*B*e))*x^5)/5 + (c*(B*c*d + 2*b*B*e + A*c*e)*x^6)/6 + (B*c^2*e*x^7)/ 7
Time = 0.45 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (a^2 A d+x^4 \left (c (2 a B e+A c d)+2 b c (A e+B d)+b^2 B e\right )+x^3 \left (2 b (a B e+A c d)+2 a c (A e+B d)+b^2 (A e+B d)\right )+x^2 \left (A \left (2 a b e+2 a c d+b^2 d\right )+a B (a e+2 b d)\right )+a x (a A e+a B d+2 A b d)+c x^5 (A c e+2 b B e+B c d)+B c^2 e x^6\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 A d x+\frac {1}{5} x^5 \left (c (2 a B e+A c d)+2 b c (A e+B d)+b^2 B e\right )+\frac {1}{4} x^4 \left (2 b (a B e+A c d)+2 a c (A e+B d)+b^2 (A e+B d)\right )+\frac {1}{3} x^3 \left (A \left (2 a b e+2 a c d+b^2 d\right )+a B (a e+2 b d)\right )+\frac {1}{2} a x^2 (a A e+a B d+2 A b d)+\frac {1}{6} c x^6 (A c e+2 b B e+B c d)+\frac {1}{7} B c^2 e x^7\) |
a^2*A*d*x + (a*(2*A*b*d + a*B*d + a*A*e)*x^2)/2 + ((a*B*(2*b*d + a*e) + A* (b^2*d + 2*a*c*d + 2*a*b*e))*x^3)/3 + ((b^2*(B*d + A*e) + 2*a*c*(B*d + A*e ) + 2*b*(A*c*d + a*B*e))*x^4)/4 + ((b^2*B*e + 2*b*c*(B*d + A*e) + c*(A*c*d + 2*a*B*e))*x^5)/5 + (c*(B*c*d + 2*b*B*e + A*c*e)*x^6)/6 + (B*c^2*e*x^7)/ 7
3.24.22.3.1 Defintions of rubi rules used
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 = 0.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(\frac {B \,c^{2} e \,x^{7}}{7}+\frac {\left (\left (A e +B d \right ) c^{2}+2 B e b c \right ) x^{6}}{6}+\frac {\left (A \,c^{2} d +2 b c \left (A e +B d \right )+B e \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 A b c d +\left (A e +B d \right ) \left (2 a c +b^{2}\right )+2 B e b a \right ) x^{4}}{4}+\frac {\left (d A \left (2 a c +b^{2}\right )+2 \left (A e +B d \right ) b a +B e \,a^{2}\right ) x^{3}}{3}+\frac {\left (2 A a b d +\left (A e +B d \right ) a^{2}\right ) x^{2}}{2}+a^{2} A d x\) | \(167\) |
| norman | \(\frac {B \,c^{2} e \,x^{7}}{7}+\left (\frac {1}{6} A \,c^{2} e +\frac {1}{3} B e b c +\frac {1}{6} B \,c^{2} d \right ) x^{6}+\left (\frac {2}{5} A b c e +\frac {1}{5} A \,c^{2} d +\frac {2}{5} B a c e +\frac {1}{5} b^{2} B e +\frac {2}{5} B b c d \right ) x^{5}+\left (\frac {1}{2} a A c e +\frac {1}{4} A \,b^{2} e +\frac {1}{2} A b c d +\frac {1}{2} B e b a +\frac {1}{2} B a c d +\frac {1}{4} B \,b^{2} d \right ) x^{4}+\left (\frac {2}{3} A a b e +\frac {2}{3} d A a c +\frac {1}{3} A d \,b^{2}+\frac {1}{3} B e \,a^{2}+\frac {2}{3} B a b d \right ) x^{3}+\left (\frac {1}{2} A \,a^{2} e +A a b d +\frac {1}{2} B \,a^{2} d \right ) x^{2}+a^{2} A d x\) | \(185\) |
| gosper | \(\frac {1}{7} B \,c^{2} e \,x^{7}+\frac {1}{6} x^{6} A \,c^{2} e +\frac {1}{3} x^{6} B e b c +\frac {1}{6} x^{6} B \,c^{2} d +\frac {2}{5} x^{5} A b c e +\frac {1}{5} x^{5} A \,c^{2} d +\frac {2}{5} x^{5} B a c e +\frac {1}{5} b^{2} B e \,x^{5}+\frac {2}{5} x^{5} B b c d +\frac {1}{2} x^{4} a A c e +\frac {1}{4} x^{4} A \,b^{2} e +\frac {1}{2} x^{4} A b c d +\frac {1}{2} x^{4} B e b a +\frac {1}{2} x^{4} B a c d +\frac {1}{4} x^{4} B \,b^{2} d +\frac {2}{3} x^{3} A a b e +\frac {2}{3} x^{3} d A a c +\frac {1}{3} A \,b^{2} d \,x^{3}+\frac {1}{3} x^{3} B e \,a^{2}+\frac {2}{3} x^{3} B a b d +\frac {1}{2} x^{2} A \,a^{2} e +x^{2} A a b d +\frac {1}{2} x^{2} B \,a^{2} d +a^{2} A d x\) | \(226\) |
| risch | \(\frac {1}{7} B \,c^{2} e \,x^{7}+\frac {1}{6} x^{6} A \,c^{2} e +\frac {1}{3} x^{6} B e b c +\frac {1}{6} x^{6} B \,c^{2} d +\frac {2}{5} x^{5} A b c e +\frac {1}{5} x^{5} A \,c^{2} d +\frac {2}{5} x^{5} B a c e +\frac {1}{5} b^{2} B e \,x^{5}+\frac {2}{5} x^{5} B b c d +\frac {1}{2} x^{4} a A c e +\frac {1}{4} x^{4} A \,b^{2} e +\frac {1}{2} x^{4} A b c d +\frac {1}{2} x^{4} B e b a +\frac {1}{2} x^{4} B a c d +\frac {1}{4} x^{4} B \,b^{2} d +\frac {2}{3} x^{3} A a b e +\frac {2}{3} x^{3} d A a c +\frac {1}{3} A \,b^{2} d \,x^{3}+\frac {1}{3} x^{3} B e \,a^{2}+\frac {2}{3} x^{3} B a b d +\frac {1}{2} x^{2} A \,a^{2} e +x^{2} A a b d +\frac {1}{2} x^{2} B \,a^{2} d +a^{2} A d x\) | \(226\) |
| parallelrisch | \(\frac {1}{7} B \,c^{2} e \,x^{7}+\frac {1}{6} x^{6} A \,c^{2} e +\frac {1}{3} x^{6} B e b c +\frac {1}{6} x^{6} B \,c^{2} d +\frac {2}{5} x^{5} A b c e +\frac {1}{5} x^{5} A \,c^{2} d +\frac {2}{5} x^{5} B a c e +\frac {1}{5} b^{2} B e \,x^{5}+\frac {2}{5} x^{5} B b c d +\frac {1}{2} x^{4} a A c e +\frac {1}{4} x^{4} A \,b^{2} e +\frac {1}{2} x^{4} A b c d +\frac {1}{2} x^{4} B e b a +\frac {1}{2} x^{4} B a c d +\frac {1}{4} x^{4} B \,b^{2} d +\frac {2}{3} x^{3} A a b e +\frac {2}{3} x^{3} d A a c +\frac {1}{3} A \,b^{2} d \,x^{3}+\frac {1}{3} x^{3} B e \,a^{2}+\frac {2}{3} x^{3} B a b d +\frac {1}{2} x^{2} A \,a^{2} e +x^{2} A a b d +\frac {1}{2} x^{2} B \,a^{2} d +a^{2} A d x\) | \(226\) |
1/7*B*c^2*e*x^7+1/6*((A*e+B*d)*c^2+2*B*e*b*c)*x^6+1/5*(A*c^2*d+2*b*c*(A*e+ B*d)+B*e*(2*a*c+b^2))*x^5+1/4*(2*A*b*c*d+(A*e+B*d)*(2*a*c+b^2)+2*B*e*b*a)* x^4+1/3*(d*A*(2*a*c+b^2)+2*(A*e+B*d)*b*a+B*e*a^2)*x^3+1/2*(2*A*a*b*d+(A*e+ B*d)*a^2)*x^2+a^2*A*d*x
Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.02 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{7} \, B c^{2} e x^{7} + \frac {1}{6} \, {\left (B c^{2} d + {\left (2 \, B b c + A c^{2}\right )} e\right )} x^{6} + \frac {1}{5} \, {\left ({\left (2 \, B b c + A c^{2}\right )} d + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e\right )} x^{5} + A a^{2} d x + \frac {1}{4} \, {\left ({\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e\right )} x^{4} + \frac {1}{3} \, {\left ({\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d + {\left (B a^{2} + 2 \, A a b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d\right )} x^{2} \]
1/7*B*c^2*e*x^7 + 1/6*(B*c^2*d + (2*B*b*c + A*c^2)*e)*x^6 + 1/5*((2*B*b*c + A*c^2)*d + (B*b^2 + 2*(B*a + A*b)*c)*e)*x^5 + A*a^2*d*x + 1/4*((B*b^2 + 2*(B*a + A*b)*c)*d + (2*B*a*b + A*b^2 + 2*A*a*c)*e)*x^4 + 1/3*((2*B*a*b + A*b^2 + 2*A*a*c)*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 = 0.04 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.28 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=A a^{2} d x + \frac {B c^{2} e x^{7}}{7} + x^{6} \left (\frac {A c^{2} e}{6} + \frac {B b c e}{3} + \frac {B c^{2} d}{6}\right ) + x^{5} \cdot \left (\frac {2 A b c e}{5} + \frac {A c^{2} d}{5} + \frac {2 B a c e}{5} + \frac {B b^{2} e}{5} + \frac {2 B b c d}{5}\right ) + x^{4} \left (\frac {A a c e}{2} + \frac {A b^{2} e}{4} + \frac {A b c d}{2} + \frac {B a b e}{2} + \frac {B a c d}{2} + \frac {B b^{2} d}{4}\right ) + x^{3} \cdot \left (\frac {2 A a b e}{3} + \frac {2 A a c d}{3} + \frac {A b^{2} d}{3} + \frac {B a^{2} e}{3} + \frac {2 B a b d}{3}\right ) + x^{2} \left (\frac {A a^{2} e}{2} + A a b d + \frac {B a^{2} d}{2}\right ) \]
A*a**2*d*x + B*c**2*e*x**7/7 + x**6*(A*c**2*e/6 + B*b*c*e/3 + B*c**2*d/6) + x**5*(2*A*b*c*e/5 + A*c**2*d/5 + 2*B*a*c*e/5 + B*b**2*e/5 + 2*B*b*c*d/5) + x**4*(A*a*c*e/2 + A*b**2*e/4 + A*b*c*d/2 + B*a*b*e/2 + B*a*c*d/2 + B*b* *2*d/4) + x**3*(2*A*a*b*e/3 + 2*A*a*c*d/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 = 0.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.02 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{7} \, B c^{2} e x^{7} + \frac {1}{6} \, {\left (B c^{2} d + {\left (2 \, B b c + A c^{2}\right )} e\right )} x^{6} + \frac {1}{5} \, {\left ({\left (2 \, B b c + A c^{2}\right )} d + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e\right )} x^{5} + A a^{2} d x + \frac {1}{4} \, {\left ({\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e\right )} x^{4} + \frac {1}{3} \, {\left ({\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d + {\left (B a^{2} + 2 \, A a b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d\right )} x^{2} \]
1/7*B*c^2*e*x^7 + 1/6*(B*c^2*d + (2*B*b*c + A*c^2)*e)*x^6 + 1/5*((2*B*b*c + A*c^2)*d + (B*b^2 + 2*(B*a + A*b)*c)*e)*x^5 + A*a^2*d*x + 1/4*((B*b^2 + 2*(B*a + A*b)*c)*d + (2*B*a*b + A*b^2 + 2*A*a*c)*e)*x^4 + 1/3*((2*B*a*b + A*b^2 + 2*A*a*c)*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 = 0.27 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.25 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{7} \, B c^{2} e x^{7} + \frac {1}{6} \, B c^{2} d x^{6} + \frac {1}{3} \, B b c e x^{6} + \frac {1}{6} \, A c^{2} e x^{6} + \frac {2}{5} \, B b c d x^{5} + \frac {1}{5} \, A c^{2} d x^{5} + \frac {1}{5} \, B b^{2} e x^{5} + \frac {2}{5} \, B a c e x^{5} + \frac {2}{5} \, A b c e x^{5} + \frac {1}{4} \, B b^{2} d x^{4} + \frac {1}{2} \, B a c d x^{4} + \frac {1}{2} \, A b c d x^{4} + \frac {1}{2} \, B a b e x^{4} + \frac {1}{4} \, A b^{2} e x^{4} + \frac {1}{2} \, A a c e x^{4} + \frac {2}{3} \, B a b d x^{3} + \frac {1}{3} \, A b^{2} d x^{3} + \frac {2}{3} \, A a c d x^{3} + \frac {1}{3} \, B a^{2} e x^{3} + \frac {2}{3} \, A a b e x^{3} + \frac {1}{2} \, B a^{2} d x^{2} + A a b d x^{2} + \frac {1}{2} \, A a^{2} e x^{2} + A a^{2} d x \]
1/7*B*c^2*e*x^7 + 1/6*B*c^2*d*x^6 + 1/3*B*b*c*e*x^6 + 1/6*A*c^2*e*x^6 + 2/ 5*B*b*c*d*x^5 + 1/5*A*c^2*d*x^5 + 1/5*B*b^2*e*x^5 + 2/5*B*a*c*e*x^5 + 2/5* A*b*c*e*x^5 + 1/4*B*b^2*d*x^4 + 1/2*B*a*c*d*x^4 + 1/2*A*b*c*d*x^4 + 1/2*B* a*b*e*x^4 + 1/4*A*b^2*e*x^4 + 1/2*A*a*c*e*x^4 + 2/3*B*a*b*d*x^3 + 1/3*A*b^ 2*d*x^3 + 2/3*A*a*c*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 = 11.03 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.02 \[ \int (A+B x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx=x^4\,\left (\frac {A\,b^2\,e}{4}+\frac {B\,b^2\,d}{4}+\frac {A\,a\,c\,e}{2}+\frac {A\,b\,c\,d}{2}+\frac {B\,a\,b\,e}{2}+\frac {B\,a\,c\,d}{2}\right )+x^3\,\left (\frac {A\,b^2\,d}{3}+\frac {B\,a^2\,e}{3}+\frac {2\,A\,a\,b\,e}{3}+\frac {2\,A\,a\,c\,d}{3}+\frac {2\,B\,a\,b\,d}{3}\right )+x^5\,\left (\frac {A\,c^2\,d}{5}+\frac {B\,b^2\,e}{5}+\frac {2\,A\,b\,c\,e}{5}+\frac {2\,B\,a\,c\,e}{5}+\frac {2\,B\,b\,c\,d}{5}\right )+x^2\,\left (\frac {A\,a^2\,e}{2}+\frac {B\,a^2\,d}{2}+A\,a\,b\,d\right )+x^6\,\left (\frac {A\,c^2\,e}{6}+\frac {B\,c^2\,d}{6}+\frac {B\,b\,c\,e}{3}\right )+A\,a^2\,d\,x+\frac {B\,c^2\,e\,x^7}{7} \]
x^4*((A*b^2*e)/4 + (B*b^2*d)/4 + (A*a*c*e)/2 + (A*b*c*d)/2 + (B*a*b*e)/2 + (B*a*c*d)/2) + x^3*((A*b^2*d)/3 + (B*a^2*e)/3 + (2*A*a*b*e)/3 + (2*A*a*c* d)/3 + (2*B*a*b*d)/3) + x^5*((A*c^2*d)/5 + (B*b^2*e)/5 + (2*A*b*c*e)/5 + ( 2*B*a*c*e)/5 + (2*B*b*c*d)/5) + x^2*((A*a^2*e)/2 + (B*a^2*d)/2 + A*a*b*d) + x^6*((A*c^2*e)/6 + (B*c^2*d)/6 + (B*b*c*e)/3) + A*a^2*d*x + (B*c^2*e*x^7 )/7