Integrand size = 28, antiderivative size = 195 \[ \int (a+b x)^3 (c+d x) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {(b c-a d) \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (a+b x)^4}{4 b^5}+\frac {\left (b^3 (B c+A d)-2 a b^2 (c C+B d)-4 a^3 d D+3 a^2 b (C d+c D)\right ) (a+b x)^5}{5 b^5}+\frac {\left (b^2 (c C+B d)+6 a^2 d D-3 a b (C d+c D)\right ) (a+b x)^6}{6 b^5}+\frac {(b C d+b c D-4 a d D) (a+b x)^7}{7 b^5}+\frac {d D (a+b x)^8}{8 b^5} \] Output:
1/4*(-a*d+b*c)*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(b*x+a)^4/b^5+1/5*(b^3*(A*d+B *c)-2*a*b^2*(B*d+C*c)-4*a^3*d*D+3*a^2*b*(C*d+D*c))*(b*x+a)^5/b^5+1/6*(b^2* (B*d+C*c)+6*a^2*d*D-3*a*b*(C*d+D*c))*(b*x+a)^6/b^5+1/7*(C*b*d-4*D*a*d+D*b* c)*(b*x+a)^7/b^5+1/8*d*D*(b*x+a)^8/b^5
Time = 0.08 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.26 \[ \int (a+b x)^3 (c+d x) \left (A+B x+C x^2+D x^3\right ) \, dx=a^3 A c x+\frac {1}{2} a^2 (3 A b c+a B c+a A d) x^2+\frac {1}{3} a (3 A b (b c+a d)+a (3 b B c+a c C+a B d)) x^3+\frac {1}{4} \left (A b^2 (b c+3 a d)+a \left (3 b^2 B c+3 a b (c C+B d)+a^2 (C d+c D)\right )\right ) x^4+\frac {1}{5} \left (b^3 (B c+A d)+3 a b^2 (c C+B d)+a^3 d D+3 a^2 b (C d+c D)\right ) x^5+\frac {1}{6} b \left (b^2 (c C+B d)+3 a^2 d D+3 a b (C d+c D)\right ) x^6+\frac {1}{7} b^2 (b C d+b c D+3 a d D) x^7+\frac {1}{8} b^3 d D x^8 \] Input:
Integrate[(a + b*x)^3*(c + d*x)*(A + B*x + C*x^2 + D*x^3),x]
Output:
a^3*A*c*x + (a^2*(3*A*b*c + a*B*c + a*A*d)*x^2)/2 + (a*(3*A*b*(b*c + a*d) + a*(3*b*B*c + a*c*C + a*B*d))*x^3)/3 + ((A*b^2*(b*c + 3*a*d) + a*(3*b^2*B *c + 3*a*b*(c*C + B*d) + a^2*(C*d + c*D)))*x^4)/4 + ((b^3*(B*c + A*d) + 3* a*b^2*(c*C + B*d) + a^3*d*D + 3*a^2*b*(C*d + c*D))*x^5)/5 + (b*(b^2*(c*C + B*d) + 3*a^2*d*D + 3*a*b*(C*d + c*D))*x^6)/6 + (b^2*(b*C*d + b*c*D + 3*a* d*D)*x^7)/7 + (b^3*d*D*x^8)/8
Time = 0.58 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2123, 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.
| \(\displaystyle \int (a+b x)^3 (c+d x) \left (A+B x+C x^2+D x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {(a+b x)^3 (b c-a d) \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^4}+\frac {(a+b x)^5 \left (6 a^2 d D-3 a b (c D+C d)+b^2 (B d+c C)\right )}{b^4}+\frac {(a+b x)^4 \left (-4 a^3 d D+3 a^2 b (c D+C d)-2 a b^2 (B d+c C)+b^3 (A d+B c)\right )}{b^4}+\frac {(a+b x)^6 (-4 a d D+b c D+b C d)}{b^4}+\frac {d D (a+b x)^7}{b^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b x)^4 (b c-a d) \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^5}+\frac {(a+b x)^6 \left (6 a^2 d D-3 a b (c D+C d)+b^2 (B d+c C)\right )}{6 b^5}+\frac {(a+b x)^5 \left (-4 a^3 d D+3 a^2 b (c D+C d)-2 a b^2 (B d+c C)+b^3 (A d+B c)\right )}{5 b^5}+\frac {(a+b x)^7 (-4 a d D+b c D+b C d)}{7 b^5}+\frac {d D (a+b x)^8}{8 b^5}\) |
Input:
Int[(a + b*x)^3*(c + d*x)*(A + B*x + C*x^2 + D*x^3),x]
Output:
((b*c - a*d)*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(a + b*x)^4)/(4*b^5) + (( b^3*(B*c + A*d) - 2*a*b^2*(c*C + B*d) - 4*a^3*d*D + 3*a^2*b*(C*d + c*D))*( a + b*x)^5)/(5*b^5) + ((b^2*(c*C + B*d) + 6*a^2*d*D - 3*a*b*(C*d + c*D))*( a + b*x)^6)/(6*b^5) + ((b*C*d + b*c*D - 4*a*d*D)*(a + b*x)^7)/(7*b^5) + (d *D*(a + b*x)^8)/(8*b^5)
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Time = 0.28 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.39
| method | result | size |
| norman | \(\frac {b^{3} d D x^{8}}{8}+\left (\frac {1}{7} b^{3} d C +\frac {3}{7} D a \,b^{2} d +\frac {1}{7} D b^{3} c \right ) x^{7}+\left (\frac {1}{6} b^{3} B d +\frac {1}{2} C a \,b^{2} d +\frac {1}{6} C \,b^{3} c +\frac {1}{2} D a^{2} b d +\frac {1}{2} D a \,b^{2} c \right ) x^{6}+\left (\frac {1}{5} b^{3} d A +\frac {3}{5} B a \,b^{2} d +\frac {1}{5} B \,b^{3} c +\frac {3}{5} C \,a^{2} b d +\frac {3}{5} a \,b^{2} c C +\frac {1}{5} a^{3} d D+\frac {3}{5} a^{2} b c D\right ) x^{5}+\left (\frac {3}{4} A a \,b^{2} d +\frac {1}{4} A \,b^{3} c +\frac {3}{4} B \,a^{2} b d +\frac {3}{4} B a \,b^{2} c +\frac {1}{4} C \,a^{3} d +\frac {3}{4} C \,a^{2} b c +\frac {1}{4} a^{3} c D\right ) x^{4}+\left (A \,a^{2} b d +A a \,b^{2} c +\frac {1}{3} B \,a^{3} d +B \,a^{2} b c +\frac {1}{3} a^{3} c C \right ) x^{3}+\left (\frac {1}{2} A \,a^{3} d +\frac {3}{2} A \,a^{2} b c +\frac {1}{2} a^{3} c B \right ) x^{2}+a^{3} c A x\) | \(272\) |
| default | \(\frac {b^{3} d D x^{8}}{8}+\frac {\left (\left (3 a \,b^{2} d +b^{3} c \right ) D+b^{3} d C \right ) x^{7}}{7}+\frac {\left (\left (3 a^{2} b d +3 a \,b^{2} c \right ) D+\left (3 a \,b^{2} d +b^{3} c \right ) C +b^{3} B d \right ) x^{6}}{6}+\frac {\left (\left (a^{3} d +3 a^{2} b c \right ) D+\left (3 a^{2} b d +3 a \,b^{2} c \right ) C +\left (3 a \,b^{2} d +b^{3} c \right ) B +b^{3} d A \right ) x^{5}}{5}+\frac {\left (a^{3} c D+\left (a^{3} d +3 a^{2} b c \right ) C +\left (3 a^{2} b d +3 a \,b^{2} c \right ) B +\left (3 a \,b^{2} d +b^{3} c \right ) A \right ) x^{4}}{4}+\frac {\left (a^{3} c C +\left (a^{3} d +3 a^{2} b c \right ) B +\left (3 a^{2} b d +3 a \,b^{2} c \right ) A \right ) x^{3}}{3}+\frac {\left (a^{3} c B +\left (a^{3} d +3 a^{2} b c \right ) A \right ) x^{2}}{2}+a^{3} c A x\) | \(279\) |
| gosper | \(\frac {1}{7} x^{7} b^{3} d C +x^{3} B \,a^{2} b c +\frac {3}{2} x^{2} A \,a^{2} b c +\frac {3}{4} x^{4} B \,a^{2} b d +\frac {3}{4} x^{4} B a \,b^{2} c +\frac {3}{4} x^{4} C \,a^{2} b c +x^{3} A \,a^{2} b d +x^{3} A a \,b^{2} c +\frac {1}{2} x^{6} D a^{2} b d +\frac {1}{2} x^{6} D a \,b^{2} c +\frac {3}{5} x^{5} B a \,b^{2} d +\frac {3}{5} x^{5} C \,a^{2} b d +\frac {3}{5} x^{5} a \,b^{2} c C +\frac {3}{5} x^{5} a^{2} b c D+\frac {3}{4} x^{4} A a \,b^{2} d +\frac {1}{7} x^{7} D b^{3} c +\frac {3}{7} x^{7} D a \,b^{2} d +\frac {1}{2} x^{6} C a \,b^{2} d +\frac {1}{8} b^{3} d D x^{8}+a^{3} c A x +\frac {1}{4} x^{4} C \,a^{3} d +\frac {1}{4} x^{4} a^{3} c D+\frac {1}{6} x^{6} C \,b^{3} c +\frac {1}{3} x^{3} a^{3} c C +\frac {1}{2} x^{2} A \,a^{3} d +\frac {1}{5} x^{5} b^{3} d A +\frac {1}{2} x^{2} a^{3} c B +\frac {1}{5} x^{5} B \,b^{3} c +\frac {1}{5} x^{5} a^{3} d D+\frac {1}{4} x^{4} A \,b^{3} c +\frac {1}{6} x^{6} b^{3} B d +\frac {1}{3} x^{3} B \,a^{3} d\) | \(332\) |
| parallelrisch | \(\frac {1}{7} x^{7} b^{3} d C +x^{3} B \,a^{2} b c +\frac {3}{2} x^{2} A \,a^{2} b c +\frac {3}{4} x^{4} B \,a^{2} b d +\frac {3}{4} x^{4} B a \,b^{2} c +\frac {3}{4} x^{4} C \,a^{2} b c +x^{3} A \,a^{2} b d +x^{3} A a \,b^{2} c +\frac {1}{2} x^{6} D a^{2} b d +\frac {1}{2} x^{6} D a \,b^{2} c +\frac {3}{5} x^{5} B a \,b^{2} d +\frac {3}{5} x^{5} C \,a^{2} b d +\frac {3}{5} x^{5} a \,b^{2} c C +\frac {3}{5} x^{5} a^{2} b c D+\frac {3}{4} x^{4} A a \,b^{2} d +\frac {1}{7} x^{7} D b^{3} c +\frac {3}{7} x^{7} D a \,b^{2} d +\frac {1}{2} x^{6} C a \,b^{2} d +\frac {1}{8} b^{3} d D x^{8}+a^{3} c A x +\frac {1}{4} x^{4} C \,a^{3} d +\frac {1}{4} x^{4} a^{3} c D+\frac {1}{6} x^{6} C \,b^{3} c +\frac {1}{3} x^{3} a^{3} c C +\frac {1}{2} x^{2} A \,a^{3} d +\frac {1}{5} x^{5} b^{3} d A +\frac {1}{2} x^{2} a^{3} c B +\frac {1}{5} x^{5} B \,b^{3} c +\frac {1}{5} x^{5} a^{3} d D+\frac {1}{4} x^{4} A \,b^{3} c +\frac {1}{6} x^{6} b^{3} B d +\frac {1}{3} x^{3} B \,a^{3} d\) | \(332\) |
| orering | \(\frac {x \left (105 b^{3} d D x^{7}+120 C \,b^{3} d \,x^{6}+360 D a \,b^{2} d \,x^{6}+120 D b^{3} c \,x^{6}+140 B \,b^{3} d \,x^{5}+420 C a \,b^{2} d \,x^{5}+140 C \,b^{3} c \,x^{5}+420 D a^{2} b d \,x^{5}+420 D a \,b^{2} c \,x^{5}+168 A \,b^{3} d \,x^{4}+504 B a \,b^{2} d \,x^{4}+168 B \,b^{3} c \,x^{4}+504 C \,a^{2} b d \,x^{4}+504 C a \,b^{2} c \,x^{4}+168 D a^{3} d \,x^{4}+504 D a^{2} b c \,x^{4}+630 A a \,b^{2} d \,x^{3}+210 A \,b^{3} c \,x^{3}+630 B \,a^{2} b d \,x^{3}+630 B a \,b^{2} c \,x^{3}+210 C \,a^{3} d \,x^{3}+630 C \,a^{2} b c \,x^{3}+210 D a^{3} c \,x^{3}+840 A \,a^{2} b d \,x^{2}+840 A a \,b^{2} c \,x^{2}+280 B \,a^{3} d \,x^{2}+840 B \,a^{2} b c \,x^{2}+280 C \,a^{3} c \,x^{2}+420 a^{3} A d x +1260 A \,a^{2} b c x +420 B \,a^{3} c x +840 a^{3} c A \right )}{840}\) | \(332\) |
Input:
int((b*x+a)^3*(d*x+c)*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVERBOSE)
Output:
1/8*b^3*d*D*x^8+(1/7*b^3*d*C+3/7*D*a*b^2*d+1/7*D*b^3*c)*x^7+(1/6*b^3*B*d+1 /2*C*a*b^2*d+1/6*C*b^3*c+1/2*D*a^2*b*d+1/2*D*a*b^2*c)*x^6+(1/5*b^3*d*A+3/5 *B*a*b^2*d+1/5*B*b^3*c+3/5*C*a^2*b*d+3/5*a*b^2*c*C+1/5*a^3*d*D+3/5*a^2*b*c *D)*x^5+(3/4*A*a*b^2*d+1/4*A*b^3*c+3/4*B*a^2*b*d+3/4*B*a*b^2*c+1/4*C*a^3*d +3/4*C*a^2*b*c+1/4*a^3*c*D)*x^4+(A*a^2*b*d+A*a*b^2*c+1/3*B*a^3*d+B*a^2*b*c +1/3*a^3*c*C)*x^3+(1/2*A*a^3*d+3/2*A*a^2*b*c+1/2*a^3*c*B)*x^2+a^3*c*A*x
Time = 0.07 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.37 \[ \int (a+b x)^3 (c+d x) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{8} \, D b^{3} d x^{8} + \frac {1}{7} \, {\left (D b^{3} c + {\left (3 \, D a b^{2} + C b^{3}\right )} d\right )} x^{7} + \frac {1}{6} \, {\left ({\left (3 \, D a b^{2} + C b^{3}\right )} c + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d\right )} x^{6} + A a^{3} c x + \frac {1}{5} \, {\left ({\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c + {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d\right )} x^{5} + \frac {1}{4} \, {\left ({\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d\right )} x^{4} + \frac {1}{3} \, {\left ({\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (A a^{3} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} c\right )} x^{2} \] Input:
integrate((b*x+a)^3*(d*x+c)*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")
Output:
1/8*D*b^3*d*x^8 + 1/7*(D*b^3*c + (3*D*a*b^2 + C*b^3)*d)*x^7 + 1/6*((3*D*a* b^2 + C*b^3)*c + (3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d)*x^6 + A*a^3*c*x + 1/5* ((3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c + (D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^ 3)*d)*x^5 + 1/4*((D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c + (C*a^3 + 3*B* a^2*b + 3*A*a*b^2)*d)*x^4 + 1/3*((C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c + (B*a^ 3 + 3*A*a^2*b)*d)*x^3 + 1/2*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*x^2
Time = 0.03 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.71 \[ \int (a+b x)^3 (c+d x) \left (A+B x+C x^2+D x^3\right ) \, dx=A a^{3} c x + \frac {D b^{3} d x^{8}}{8} + x^{7} \left (\frac {C b^{3} d}{7} + \frac {3 D a b^{2} d}{7} + \frac {D b^{3} c}{7}\right ) + x^{6} \left (\frac {B b^{3} d}{6} + \frac {C a b^{2} d}{2} + \frac {C b^{3} c}{6} + \frac {D a^{2} b d}{2} + \frac {D a b^{2} c}{2}\right ) + x^{5} \left (\frac {A b^{3} d}{5} + \frac {3 B a b^{2} d}{5} + \frac {B b^{3} c}{5} + \frac {3 C a^{2} b d}{5} + \frac {3 C a b^{2} c}{5} + \frac {D a^{3} d}{5} + \frac {3 D a^{2} b c}{5}\right ) + x^{4} \cdot \left (\frac {3 A a b^{2} d}{4} + \frac {A b^{3} c}{4} + \frac {3 B a^{2} b d}{4} + \frac {3 B a b^{2} c}{4} + \frac {C a^{3} d}{4} + \frac {3 C a^{2} b c}{4} + \frac {D a^{3} c}{4}\right ) + x^{3} \left (A a^{2} b d + A a b^{2} c + \frac {B a^{3} d}{3} + B a^{2} b c + \frac {C a^{3} c}{3}\right ) + x^{2} \left (\frac {A a^{3} d}{2} + \frac {3 A a^{2} b c}{2} + \frac {B a^{3} c}{2}\right ) \] Input:
integrate((b*x+a)**3*(d*x+c)*(D*x**3+C*x**2+B*x+A),x)
Output:
A*a**3*c*x + D*b**3*d*x**8/8 + x**7*(C*b**3*d/7 + 3*D*a*b**2*d/7 + D*b**3* c/7) + x**6*(B*b**3*d/6 + C*a*b**2*d/2 + C*b**3*c/6 + D*a**2*b*d/2 + D*a*b **2*c/2) + x**5*(A*b**3*d/5 + 3*B*a*b**2*d/5 + B*b**3*c/5 + 3*C*a**2*b*d/5 + 3*C*a*b**2*c/5 + D*a**3*d/5 + 3*D*a**2*b*c/5) + x**4*(3*A*a*b**2*d/4 + A*b**3*c/4 + 3*B*a**2*b*d/4 + 3*B*a*b**2*c/4 + C*a**3*d/4 + 3*C*a**2*b*c/4 + D*a**3*c/4) + x**3*(A*a**2*b*d + A*a*b**2*c + B*a**3*d/3 + B*a**2*b*c + C*a**3*c/3) + x**2*(A*a**3*d/2 + 3*A*a**2*b*c/2 + B*a**3*c/2)
Time = 0.03 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.37 \[ \int (a+b x)^3 (c+d x) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{8} \, D b^{3} d x^{8} + \frac {1}{7} \, {\left (D b^{3} c + {\left (3 \, D a b^{2} + C b^{3}\right )} d\right )} x^{7} + \frac {1}{6} \, {\left ({\left (3 \, D a b^{2} + C b^{3}\right )} c + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d\right )} x^{6} + A a^{3} c x + \frac {1}{5} \, {\left ({\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c + {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d\right )} x^{5} + \frac {1}{4} \, {\left ({\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d\right )} x^{4} + \frac {1}{3} \, {\left ({\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (A a^{3} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} c\right )} x^{2} \] Input:
integrate((b*x+a)^3*(d*x+c)*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")
Output:
1/8*D*b^3*d*x^8 + 1/7*(D*b^3*c + (3*D*a*b^2 + C*b^3)*d)*x^7 + 1/6*((3*D*a* b^2 + C*b^3)*c + (3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d)*x^6 + A*a^3*c*x + 1/5* ((3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c + (D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^ 3)*d)*x^5 + 1/4*((D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c + (C*a^3 + 3*B* a^2*b + 3*A*a*b^2)*d)*x^4 + 1/3*((C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c + (B*a^ 3 + 3*A*a^2*b)*d)*x^3 + 1/2*(A*a^3*d + (B*a^3 + 3*A*a^2*b)*c)*x^2
Time = 0.12 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.70 \[ \int (a+b x)^3 (c+d x) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{8} \, D b^{3} d x^{8} + \frac {1}{7} \, D b^{3} c x^{7} + \frac {3}{7} \, D a b^{2} d x^{7} + \frac {1}{7} \, C b^{3} d x^{7} + \frac {1}{2} \, D a b^{2} c x^{6} + \frac {1}{6} \, C b^{3} c x^{6} + \frac {1}{2} \, D a^{2} b d x^{6} + \frac {1}{2} \, C a b^{2} d x^{6} + \frac {1}{6} \, B b^{3} d x^{6} + \frac {3}{5} \, D a^{2} b c x^{5} + \frac {3}{5} \, C a b^{2} c x^{5} + \frac {1}{5} \, B b^{3} c x^{5} + \frac {1}{5} \, D a^{3} d x^{5} + \frac {3}{5} \, C a^{2} b d x^{5} + \frac {3}{5} \, B a b^{2} d x^{5} + \frac {1}{5} \, A b^{3} d x^{5} + \frac {1}{4} \, D a^{3} c x^{4} + \frac {3}{4} \, C a^{2} b c x^{4} + \frac {3}{4} \, B a b^{2} c x^{4} + \frac {1}{4} \, A b^{3} c x^{4} + \frac {1}{4} \, C a^{3} d x^{4} + \frac {3}{4} \, B a^{2} b d x^{4} + \frac {3}{4} \, A a b^{2} d x^{4} + \frac {1}{3} \, C a^{3} c x^{3} + B a^{2} b c x^{3} + A a b^{2} c x^{3} + \frac {1}{3} \, B a^{3} d x^{3} + A a^{2} b d x^{3} + \frac {1}{2} \, B a^{3} c x^{2} + \frac {3}{2} \, A a^{2} b c x^{2} + \frac {1}{2} \, A a^{3} d x^{2} + A a^{3} c x \] Input:
integrate((b*x+a)^3*(d*x+c)*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")
Output:
1/8*D*b^3*d*x^8 + 1/7*D*b^3*c*x^7 + 3/7*D*a*b^2*d*x^7 + 1/7*C*b^3*d*x^7 + 1/2*D*a*b^2*c*x^6 + 1/6*C*b^3*c*x^6 + 1/2*D*a^2*b*d*x^6 + 1/2*C*a*b^2*d*x^ 6 + 1/6*B*b^3*d*x^6 + 3/5*D*a^2*b*c*x^5 + 3/5*C*a*b^2*c*x^5 + 1/5*B*b^3*c* x^5 + 1/5*D*a^3*d*x^5 + 3/5*C*a^2*b*d*x^5 + 3/5*B*a*b^2*d*x^5 + 1/5*A*b^3* d*x^5 + 1/4*D*a^3*c*x^4 + 3/4*C*a^2*b*c*x^4 + 3/4*B*a*b^2*c*x^4 + 1/4*A*b^ 3*c*x^4 + 1/4*C*a^3*d*x^4 + 3/4*B*a^2*b*d*x^4 + 3/4*A*a*b^2*d*x^4 + 1/3*C* a^3*c*x^3 + B*a^2*b*c*x^3 + A*a*b^2*c*x^3 + 1/3*B*a^3*d*x^3 + A*a^2*b*d*x^ 3 + 1/2*B*a^3*c*x^2 + 3/2*A*a^2*b*c*x^2 + 1/2*A*a^3*d*x^2 + A*a^3*c*x
Time = 4.81 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.70 \[ \int (a+b x)^3 (c+d x) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {a^3\,c\,x^4\,D}{4}+\frac {a^3\,d\,x^5\,D}{5}+\frac {b^3\,c\,x^7\,D}{7}+\frac {b^3\,d\,x^8\,D}{8}+A\,a^3\,c\,x+\frac {A\,a^3\,d\,x^2}{2}+\frac {B\,a^3\,c\,x^2}{2}+\frac {A\,b^3\,c\,x^4}{4}+\frac {B\,a^3\,d\,x^3}{3}+\frac {C\,a^3\,c\,x^3}{3}+\frac {A\,b^3\,d\,x^5}{5}+\frac {B\,b^3\,c\,x^5}{5}+\frac {C\,a^3\,d\,x^4}{4}+\frac {B\,b^3\,d\,x^6}{6}+\frac {C\,b^3\,c\,x^6}{6}+\frac {C\,b^3\,d\,x^7}{7}+\frac {3\,C\,a^2\,b\,d\,x^5}{5}+\frac {C\,a\,b^2\,d\,x^6}{2}+\frac {3\,a^2\,b\,c\,x^5\,D}{5}+\frac {a\,b^2\,c\,x^6\,D}{2}+\frac {a^2\,b\,d\,x^6\,D}{2}+\frac {3\,a\,b^2\,d\,x^7\,D}{7}+\frac {3\,A\,a^2\,b\,c\,x^2}{2}+A\,a\,b^2\,c\,x^3+A\,a^2\,b\,d\,x^3+B\,a^2\,b\,c\,x^3+\frac {3\,A\,a\,b^2\,d\,x^4}{4}+\frac {3\,B\,a\,b^2\,c\,x^4}{4}+\frac {3\,B\,a^2\,b\,d\,x^4}{4}+\frac {3\,C\,a^2\,b\,c\,x^4}{4}+\frac {3\,B\,a\,b^2\,d\,x^5}{5}+\frac {3\,C\,a\,b^2\,c\,x^5}{5} \] Input:
int((a + b*x)^3*(c + d*x)*(A + B*x + C*x^2 + x^3*D),x)
Output:
(a^3*c*x^4*D)/4 + (a^3*d*x^5*D)/5 + (b^3*c*x^7*D)/7 + (b^3*d*x^8*D)/8 + A* a^3*c*x + (A*a^3*d*x^2)/2 + (B*a^3*c*x^2)/2 + (A*b^3*c*x^4)/4 + (B*a^3*d*x ^3)/3 + (C*a^3*c*x^3)/3 + (A*b^3*d*x^5)/5 + (B*b^3*c*x^5)/5 + (C*a^3*d*x^4 )/4 + (B*b^3*d*x^6)/6 + (C*b^3*c*x^6)/6 + (C*b^3*d*x^7)/7 + (3*C*a^2*b*d*x ^5)/5 + (C*a*b^2*d*x^6)/2 + (3*a^2*b*c*x^5*D)/5 + (a*b^2*c*x^6*D)/2 + (a^2 *b*d*x^6*D)/2 + (3*a*b^2*d*x^7*D)/7 + (3*A*a^2*b*c*x^2)/2 + A*a*b^2*c*x^3 + A*a^2*b*d*x^3 + B*a^2*b*c*x^3 + (3*A*a*b^2*d*x^4)/4 + (3*B*a*b^2*c*x^4)/ 4 + (3*B*a^2*b*d*x^4)/4 + (3*C*a^2*b*c*x^4)/4 + (3*B*a*b^2*d*x^5)/5 + (3*C *a*b^2*c*x^5)/5
Time = 0.14 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.18 \[ \int (a+b x)^3 (c+d x) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {x \left (105 b^{3} d^{2} x^{7}+360 a \,b^{2} d^{2} x^{6}+240 b^{3} c d \,x^{6}+420 a^{2} b \,d^{2} x^{5}+840 a \,b^{2} c d \,x^{5}+140 b^{4} d \,x^{5}+140 b^{3} c^{2} x^{5}+168 a^{3} d^{2} x^{4}+1008 a^{2} b c d \,x^{4}+672 a \,b^{3} d \,x^{4}+504 a \,b^{2} c^{2} x^{4}+168 b^{4} c \,x^{4}+420 a^{3} c d \,x^{3}+1260 a^{2} b^{2} d \,x^{3}+630 a^{2} b \,c^{2} x^{3}+840 a \,b^{3} c \,x^{3}+1120 a^{3} b d \,x^{2}+280 a^{3} c^{2} x^{2}+1680 a^{2} b^{2} c \,x^{2}+420 a^{4} d x +1680 a^{3} b c x +840 a^{4} c \right )}{840} \] Input:
int((b*x+a)^3*(d*x+c)*(D*x^3+C*x^2+B*x+A),x)
Output:
(x*(840*a**4*c + 420*a**4*d*x + 1680*a**3*b*c*x + 1120*a**3*b*d*x**2 + 280 *a**3*c**2*x**2 + 420*a**3*c*d*x**3 + 168*a**3*d**2*x**4 + 1680*a**2*b**2* c*x**2 + 1260*a**2*b**2*d*x**3 + 630*a**2*b*c**2*x**3 + 1008*a**2*b*c*d*x* *4 + 420*a**2*b*d**2*x**5 + 840*a*b**3*c*x**3 + 672*a*b**3*d*x**4 + 504*a* b**2*c**2*x**4 + 840*a*b**2*c*d*x**5 + 360*a*b**2*d**2*x**6 + 168*b**4*c*x **4 + 140*b**4*d*x**5 + 140*b**3*c**2*x**5 + 240*b**3*c*d*x**6 + 105*b**3* d**2*x**7))/840