Integrand size = 41, antiderivative size = 112 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right ) \, dx=a A d x+\frac {1}{2} (A b d+a B d+a A e) x^2+\frac {1}{3} (b B d+A c d+a C d+A b e+a B e) x^3+\frac {1}{4} (B c d+b C d+b B e+A c e+a C e) x^4+\frac {1}{5} (c C d+B c e+b C e) x^5+\frac {1}{6} c C e x^6 \] Output:
a*A*d*x+1/2*(A*a*e+A*b*d+B*a*d)*x^2+1/3*(A*b*e+A*c*d+B*a*e+B*b*d+C*a*d)*x^ 3+1/4*(A*c*e+B*b*e+B*c*d+C*a*e+C*b*d)*x^4+1/5*(B*c*e+C*b*e+C*c*d)*x^5+1/6* c*C*e*x^6
Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right ) \, dx=a A d x+\frac {1}{2} (A b d+a B d+a A e) x^2+\frac {1}{3} (b B d+A c d+a C d+A b e+a B e) x^3+\frac {1}{4} (B c d+b C d+b B e+A c e+a C e) x^4+\frac {1}{5} (c C d+B c e+b C e) x^5+\frac {1}{6} c C e x^6 \] Input:
Integrate[(A + B*x + C*x^2)*(a*d + (b*d + a*e)*x + (c*d + b*e)*x^2 + c*e*x ^3),x]
Output:
a*A*d*x + ((A*b*d + a*B*d + a*A*e)*x^2)/2 + ((b*B*d + A*c*d + a*C*d + A*b* e + a*B*e)*x^3)/3 + ((B*c*d + b*C*d + b*B*e + A*c*e + a*C*e)*x^4)/4 + ((c* C*d + B*c*e + b*C*e)*x^5)/5 + (c*C*e*x^6)/6
Time = 0.52 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {2188, 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.
\(\displaystyle \int \left (A+B x+C x^2\right ) \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right ) \, dx\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \int \left (x^3 (a C e+A c e+b B e+b C d+B c d)+x^2 (a B e+a C d+A b e+A c d+b B d)+x (a A e+a B d+A b d)+a A d+x^4 (b C e+B c e+c C d)+c C e x^5\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} x^4 (a C e+A c e+b B e+b C d+B c d)+\frac {1}{3} x^3 (a B e+a C d+A b e+A c d+b B d)+\frac {1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac {1}{5} x^5 (b C e+B c e+c C d)+\frac {1}{6} c C e x^6\) |
Input:
Int[(A + B*x + C*x^2)*(a*d + (b*d + a*e)*x + (c*d + b*e)*x^2 + c*e*x^3),x]
Output:
a*A*d*x + ((A*b*d + a*B*d + a*A*e)*x^2)/2 + ((b*B*d + A*c*d + a*C*d + A*b* e + a*B*e)*x^3)/3 + ((B*c*d + b*C*d + b*B*e + A*c*e + a*C*e)*x^4)/4 + ((c* C*d + B*c*e + b*C*e)*x^5)/5 + (c*C*e*x^6)/6
Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq , x] && IGtQ[p, -2]
Time = 0.18 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {c C e \,x^{6}}{6}+\frac {\left (\left (e b +c d \right ) C +B c e \right ) x^{5}}{5}+\frac {\left (\left (a e +b d \right ) C +\left (e b +c d \right ) B +A c e \right ) x^{4}}{4}+\frac {\left (C a d +\left (a e +b d \right ) B +\left (e b +c d \right ) A \right ) x^{3}}{3}+\frac {\left (B a d +A \left (a e +b d \right )\right ) x^{2}}{2}+a A d x\) | \(109\) |
norman | \(\frac {c C e \,x^{6}}{6}+\left (\frac {1}{5} B c e +\frac {1}{5} C b e +\frac {1}{5} C c d \right ) x^{5}+\left (\frac {1}{4} A c e +\frac {1}{4} B b e +\frac {1}{4} B c d +\frac {1}{4} C a e +\frac {1}{4} C b d \right ) x^{4}+\left (\frac {1}{3} A b e +\frac {1}{3} A c d +\frac {1}{3} B a e +\frac {1}{3} B b d +\frac {1}{3} C a d \right ) x^{3}+\left (\frac {1}{2} A a e +\frac {1}{2} A b d +\frac {1}{2} B a d \right ) x^{2}+a A d x\) | \(115\) |
gosper | \(\frac {x \left (10 C c e \,x^{5}+12 B c e \,x^{4}+12 C b e \,x^{4}+12 C c d \,x^{4}+15 A c e \,x^{3}+15 B b e \,x^{3}+15 B c d \,x^{3}+15 C a e \,x^{3}+15 C b d \,x^{3}+20 A b e \,x^{2}+20 A c d \,x^{2}+20 B a e \,x^{2}+20 B b d \,x^{2}+20 C a d \,x^{2}+30 A a e x +30 A b d x +30 B a d x +60 a d A \right )}{60}\) | \(140\) |
risch | \(\frac {1}{6} c C e \,x^{6}+\frac {1}{5} x^{5} B c e +\frac {1}{5} x^{5} C b e +\frac {1}{5} x^{5} C c d +\frac {1}{4} A c e \,x^{4}+\frac {1}{4} B b e \,x^{4}+\frac {1}{4} B c d \,x^{4}+\frac {1}{4} x^{4} C a e +\frac {1}{4} x^{4} C b d +\frac {1}{3} x^{3} A b e +\frac {1}{3} x^{3} A c d +\frac {1}{3} x^{3} B a e +\frac {1}{3} x^{3} B b d +\frac {1}{3} x^{3} C a d +\frac {1}{2} x^{2} A a e +\frac {1}{2} x^{2} A b d +\frac {1}{2} x^{2} B a d +a A d x\) | \(143\) |
parallelrisch | \(\frac {1}{6} c C e \,x^{6}+\frac {1}{5} x^{5} B c e +\frac {1}{5} x^{5} C b e +\frac {1}{5} x^{5} C c d +\frac {1}{4} A c e \,x^{4}+\frac {1}{4} B b e \,x^{4}+\frac {1}{4} B c d \,x^{4}+\frac {1}{4} x^{4} C a e +\frac {1}{4} x^{4} C b d +\frac {1}{3} x^{3} A b e +\frac {1}{3} x^{3} A c d +\frac {1}{3} x^{3} B a e +\frac {1}{3} x^{3} B b d +\frac {1}{3} x^{3} C a d +\frac {1}{2} x^{2} A a e +\frac {1}{2} x^{2} A b d +\frac {1}{2} x^{2} B a d +a A d x\) | \(143\) |
orering | \(\frac {x \left (10 C c e \,x^{5}+12 B c e \,x^{4}+12 C b e \,x^{4}+12 C c d \,x^{4}+15 A c e \,x^{3}+15 B b e \,x^{3}+15 B c d \,x^{3}+15 C a e \,x^{3}+15 C b d \,x^{3}+20 A b e \,x^{2}+20 A c d \,x^{2}+20 B a e \,x^{2}+20 B b d \,x^{2}+20 C a d \,x^{2}+30 A a e x +30 A b d x +30 B a d x +60 a d A \right ) \left (a d +\left (a e +b d \right ) x +\left (e b +c d \right ) x^{2}+c e \,x^{3}\right )}{60 \left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\) | \(189\) |
Input:
int((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3),x,method=_RETURN VERBOSE)
Output:
1/6*c*C*e*x^6+1/5*((b*e+c*d)*C+B*c*e)*x^5+1/4*((a*e+b*d)*C+(b*e+c*d)*B+A*c *e)*x^4+1/3*(C*a*d+(a*e+b*d)*B+(b*e+c*d)*A)*x^3+1/2*(B*a*d+A*(a*e+b*d))*x^ 2+a*A*d*x
Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right ) \, dx=\frac {1}{6} \, C c e x^{6} + \frac {1}{5} \, {\left (C c d + {\left (C b + B c\right )} e\right )} x^{5} + \frac {1}{4} \, {\left ({\left (C b + B c\right )} d + {\left (C a + B b + A c\right )} e\right )} x^{4} + A a d x + \frac {1}{3} \, {\left ({\left (C a + B b + A c\right )} d + {\left (B a + A b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a e + {\left (B a + A b\right )} d\right )} x^{2} \] Input:
integrate((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3),x, algorit hm="fricas")
Output:
1/6*C*c*e*x^6 + 1/5*(C*c*d + (C*b + B*c)*e)*x^5 + 1/4*((C*b + B*c)*d + (C* a + B*b + A*c)*e)*x^4 + A*a*d*x + 1/3*((C*a + B*b + A*c)*d + (B*a + A*b)*e )*x^3 + 1/2*(A*a*e + (B*a + A*b)*d)*x^2
Time = 0.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.23 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right ) \, dx=A a d x + \frac {C c e x^{6}}{6} + x^{5} \left (\frac {B c e}{5} + \frac {C b e}{5} + \frac {C c d}{5}\right ) + x^{4} \left (\frac {A c e}{4} + \frac {B b e}{4} + \frac {B c d}{4} + \frac {C a e}{4} + \frac {C b d}{4}\right ) + x^{3} \left (\frac {A b e}{3} + \frac {A c d}{3} + \frac {B a e}{3} + \frac {B b d}{3} + \frac {C a d}{3}\right ) + x^{2} \left (\frac {A a e}{2} + \frac {A b d}{2} + \frac {B a d}{2}\right ) \] Input:
integrate((C*x**2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x**2+c*e*x**3),x)
Output:
A*a*d*x + C*c*e*x**6/6 + x**5*(B*c*e/5 + C*b*e/5 + C*c*d/5) + x**4*(A*c*e/ 4 + B*b*e/4 + B*c*d/4 + C*a*e/4 + C*b*d/4) + x**3*(A*b*e/3 + A*c*d/3 + B*a *e/3 + B*b*d/3 + C*a*d/3) + x**2*(A*a*e/2 + A*b*d/2 + B*a*d/2)
Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right ) \, dx=\frac {1}{6} \, C c e x^{6} + \frac {1}{5} \, {\left (C c d + {\left (C b + B c\right )} e\right )} x^{5} + \frac {1}{4} \, {\left ({\left (C b + B c\right )} d + {\left (C a + B b + A c\right )} e\right )} x^{4} + A a d x + \frac {1}{3} \, {\left ({\left (C a + B b + A c\right )} d + {\left (B a + A b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a e + {\left (B a + A b\right )} d\right )} x^{2} \] Input:
integrate((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3),x, algorit hm="maxima")
Output:
1/6*C*c*e*x^6 + 1/5*(C*c*d + (C*b + B*c)*e)*x^5 + 1/4*((C*b + B*c)*d + (C* a + B*b + A*c)*e)*x^4 + A*a*d*x + 1/3*((C*a + B*b + A*c)*d + (B*a + A*b)*e )*x^3 + 1/2*(A*a*e + (B*a + A*b)*d)*x^2
Time = 0.11 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.27 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right ) \, dx=\frac {1}{6} \, C c e x^{6} + \frac {1}{5} \, C c d x^{5} + \frac {1}{5} \, C b e x^{5} + \frac {1}{5} \, B c e x^{5} + \frac {1}{4} \, C b d x^{4} + \frac {1}{4} \, B c d x^{4} + \frac {1}{4} \, C a e x^{4} + \frac {1}{4} \, B b e x^{4} + \frac {1}{4} \, A c e x^{4} + \frac {1}{3} \, C a d x^{3} + \frac {1}{3} \, B b d x^{3} + \frac {1}{3} \, A c d x^{3} + \frac {1}{3} \, B a e x^{3} + \frac {1}{3} \, A b e x^{3} + \frac {1}{2} \, B a d x^{2} + \frac {1}{2} \, A b d x^{2} + \frac {1}{2} \, A a e x^{2} + A a d x \] Input:
integrate((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3),x, algorit hm="giac")
Output:
1/6*C*c*e*x^6 + 1/5*C*c*d*x^5 + 1/5*C*b*e*x^5 + 1/5*B*c*e*x^5 + 1/4*C*b*d* x^4 + 1/4*B*c*d*x^4 + 1/4*C*a*e*x^4 + 1/4*B*b*e*x^4 + 1/4*A*c*e*x^4 + 1/3* C*a*d*x^3 + 1/3*B*b*d*x^3 + 1/3*A*c*d*x^3 + 1/3*B*a*e*x^3 + 1/3*A*b*e*x^3 + 1/2*B*a*d*x^2 + 1/2*A*b*d*x^2 + 1/2*A*a*e*x^2 + A*a*d*x
Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right ) \, dx=\frac {C\,c\,e\,x^6}{6}+\left (\frac {B\,c\,e}{5}+\frac {C\,b\,e}{5}+\frac {C\,c\,d}{5}\right )\,x^5+\left (\frac {A\,c\,e}{4}+\frac {B\,b\,e}{4}+\frac {B\,c\,d}{4}+\frac {C\,a\,e}{4}+\frac {C\,b\,d}{4}\right )\,x^4+\left (\frac {A\,b\,e}{3}+\frac {A\,c\,d}{3}+\frac {B\,a\,e}{3}+\frac {B\,b\,d}{3}+\frac {C\,a\,d}{3}\right )\,x^3+\left (\frac {A\,a\,e}{2}+\frac {A\,b\,d}{2}+\frac {B\,a\,d}{2}\right )\,x^2+A\,a\,d\,x \] Input:
int((A + B*x + C*x^2)*(a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c*e*x^3),x)
Output:
x^3*((A*b*e)/3 + (A*c*d)/3 + (B*a*e)/3 + (B*b*d)/3 + (C*a*d)/3) + x^4*((A* c*e)/4 + (B*b*e)/4 + (B*c*d)/4 + (C*a*e)/4 + (C*b*d)/4) + x^2*((A*a*e)/2 + (A*b*d)/2 + (B*a*d)/2) + x^5*((B*c*e)/5 + (C*b*e)/5 + (C*c*d)/5) + (C*c*e *x^6)/6 + A*a*d*x
Time = 0.16 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \left (A+B x+C x^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right ) \, dx=\frac {x \left (10 c^{2} e \,x^{5}+24 b c e \,x^{4}+12 c^{2} d \,x^{4}+30 a c e \,x^{3}+15 b^{2} e \,x^{3}+30 b c d \,x^{3}+40 a b e \,x^{2}+40 a c d \,x^{2}+20 b^{2} d \,x^{2}+30 a^{2} e x +60 a b d x +60 a^{2} d \right )}{60} \] Input:
int((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3),x)
Output:
(x*(60*a**2*d + 30*a**2*e*x + 60*a*b*d*x + 40*a*b*e*x**2 + 40*a*c*d*x**2 + 30*a*c*e*x**3 + 20*b**2*d*x**2 + 15*b**2*e*x**3 + 30*b*c*d*x**3 + 24*b*c* e*x**4 + 12*c**2*d*x**4 + 10*c**2*e*x**5))/60