Integrand size = 23, antiderivative size = 133 \[ \int (A+B x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{4 e^4}+\frac {\left (3 B c d^2-B e (2 b d-a e)-A e (2 c d-b e)\right ) (d+e x)^5}{5 e^4}-\frac {(3 B c d-b B e-A c e) (d+e x)^6}{6 e^4}+\frac {B c (d+e x)^7}{7 e^4} \] Output:
-1/4*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)*(e*x+d)^4/e^4+1/5*(3*B*c*d^2-B*e*(-a*e +2*b*d)-A*e*(-b*e+2*c*d))*(e*x+d)^5/e^4-1/6*(-A*c*e-B*b*e+3*B*c*d)*(e*x+d) ^6/e^4+1/7*B*c*(e*x+d)^7/e^4
Time = 0.09 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.44 \[ \int (A+B x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=a A d^3 x+\frac {1}{2} d^2 (A b d+a B d+3 a A e) x^2+\frac {1}{3} d \left (A c d^2+3 a B d e+3 a A e^2+b d (B d+3 A e)\right ) x^3+\frac {1}{4} \left (B \left (c d^3+3 d e (b d+a e)\right )+A e \left (3 c d^2+e (3 b d+a e)\right )\right ) x^4+\frac {1}{5} e \left (3 B c d^2+B e (3 b d+a e)+A e (3 c d+b e)\right ) x^5+\frac {1}{6} e^2 (3 B c d+b B e+A c e) x^6+\frac {1}{7} B c e^3 x^7 \] Input:
Integrate[(A + B*x)*(d + e*x)^3*(a + b*x + c*x^2),x]
Output:
a*A*d^3*x + (d^2*(A*b*d + a*B*d + 3*a*A*e)*x^2)/2 + (d*(A*c*d^2 + 3*a*B*d* e + 3*a*A*e^2 + b*d*(B*d + 3*A*e))*x^3)/3 + ((B*(c*d^3 + 3*d*e*(b*d + a*e) ) + A*e*(3*c*d^2 + e*(3*b*d + a*e)))*x^4)/4 + (e*(3*B*c*d^2 + B*e*(3*b*d + a*e) + A*e*(3*c*d + b*e))*x^5)/5 + (e^2*(3*B*c*d + b*B*e + A*c*e)*x^6)/6 + (B*c*e^3*x^7)/7
Time = 0.61 (sec) , antiderivative size = 133, 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.
| \(\displaystyle \int (A+B x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^4 \left (-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2\right )}{e^3}+\frac {(d+e x)^3 (A e-B d) \left (a e^2-b d e+c d^2\right )}{e^3}+\frac {(d+e x)^5 (A c e+b B e-3 B c d)}{e^3}+\frac {B c (d+e x)^6}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^5 \left (-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2\right )}{5 e^4}-\frac {(d+e x)^4 (B d-A e) \left (a e^2-b d e+c d^2\right )}{4 e^4}-\frac {(d+e x)^6 (-A c e-b B e+3 B c d)}{6 e^4}+\frac {B c (d+e x)^7}{7 e^4}\) |
Input:
Int[(A + B*x)*(d + e*x)^3*(a + b*x + c*x^2),x]
Output:
-1/4*((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4)/e^4 + ((3*B*c*d^2 - B*e*(2*b*d - a*e) - A*e*(2*c*d - b*e))*(d + e*x)^5)/(5*e^4) - ((3*B*c*d - b*B*e - A*c*e)*(d + e*x)^6)/(6*e^4) + (B*c*(d + e*x)^7)/(7*e^4)
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 = 0.70 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.56
| method | result | size |
| norman | \(\frac {B \,e^{3} c \,x^{7}}{7}+\left (\frac {1}{6} A c \,e^{3}+\frac {1}{6} B \,e^{3} b +\frac {1}{2} B c d \,e^{2}\right ) x^{6}+\left (\frac {1}{5} A b \,e^{3}+\frac {3}{5} A c d \,e^{2}+\frac {1}{5} B \,e^{3} a +\frac {3}{5} B b d \,e^{2}+\frac {3}{5} B c \,d^{2} e \right ) x^{5}+\left (\frac {1}{4} A a \,e^{3}+\frac {3}{4} A b d \,e^{2}+\frac {3}{4} A c \,d^{2} e +\frac {3}{4} B a d \,e^{2}+\frac {3}{4} B b \,d^{2} e +\frac {1}{4} B c \,d^{3}\right ) x^{4}+\left (A a d \,e^{2}+A b \,d^{2} e +\frac {1}{3} A c \,d^{3}+B a \,d^{2} e +\frac {1}{3} B b \,d^{3}\right ) x^{3}+\left (\frac {3}{2} A a \,d^{2} e +\frac {1}{2} A b \,d^{3}+\frac {1}{2} B a \,d^{3}\right ) x^{2}+A \,d^{3} a x\) | \(207\) |
| default | \(\frac {B \,e^{3} c \,x^{7}}{7}+\frac {\left (\left (A \,e^{3}+3 B d \,e^{2}\right ) c +B \,e^{3} b \right ) x^{6}}{6}+\frac {\left (\left (3 A d \,e^{2}+3 e B \,d^{2}\right ) c +\left (A \,e^{3}+3 B d \,e^{2}\right ) b +B \,e^{3} a \right ) x^{5}}{5}+\frac {\left (\left (3 A \,d^{2} e +B \,d^{3}\right ) c +\left (3 A d \,e^{2}+3 e B \,d^{2}\right ) b +\left (A \,e^{3}+3 B d \,e^{2}\right ) a \right ) x^{4}}{4}+\frac {\left (A c \,d^{3}+\left (3 A \,d^{2} e +B \,d^{3}\right ) b +\left (3 A d \,e^{2}+3 e B \,d^{2}\right ) a \right ) x^{3}}{3}+\frac {\left (A b \,d^{3}+\left (3 A \,d^{2} e +B \,d^{3}\right ) a \right ) x^{2}}{2}+A \,d^{3} a x\) | \(214\) |
| gosper | \(\frac {1}{7} B \,e^{3} c \,x^{7}+\frac {1}{6} x^{6} A c \,e^{3}+\frac {1}{6} x^{6} B \,e^{3} b +\frac {1}{2} x^{6} B c d \,e^{2}+\frac {1}{5} x^{5} A b \,e^{3}+\frac {3}{5} x^{5} A c d \,e^{2}+\frac {1}{5} x^{5} B \,e^{3} a +\frac {3}{5} x^{5} B b d \,e^{2}+\frac {3}{5} x^{5} B c \,d^{2} e +\frac {1}{4} x^{4} A a \,e^{3}+\frac {3}{4} x^{4} A b d \,e^{2}+\frac {3}{4} x^{4} A c \,d^{2} e +\frac {3}{4} x^{4} B a d \,e^{2}+\frac {3}{4} x^{4} B b \,d^{2} e +\frac {1}{4} x^{4} B c \,d^{3}+x^{3} A a d \,e^{2}+x^{3} A b \,d^{2} e +\frac {1}{3} x^{3} A c \,d^{3}+x^{3} B a \,d^{2} e +\frac {1}{3} B b \,d^{3} x^{3}+\frac {3}{2} x^{2} A a \,d^{2} e +\frac {1}{2} A b \,d^{3} x^{2}+\frac {1}{2} x^{2} B a \,d^{3}+A \,d^{3} a x\) | \(248\) |
| risch | \(\frac {1}{7} B \,e^{3} c \,x^{7}+\frac {1}{6} x^{6} A c \,e^{3}+\frac {1}{6} x^{6} B \,e^{3} b +\frac {1}{2} x^{6} B c d \,e^{2}+\frac {1}{5} x^{5} A b \,e^{3}+\frac {3}{5} x^{5} A c d \,e^{2}+\frac {1}{5} x^{5} B \,e^{3} a +\frac {3}{5} x^{5} B b d \,e^{2}+\frac {3}{5} x^{5} B c \,d^{2} e +\frac {1}{4} x^{4} A a \,e^{3}+\frac {3}{4} x^{4} A b d \,e^{2}+\frac {3}{4} x^{4} A c \,d^{2} e +\frac {3}{4} x^{4} B a d \,e^{2}+\frac {3}{4} x^{4} B b \,d^{2} e +\frac {1}{4} x^{4} B c \,d^{3}+x^{3} A a d \,e^{2}+x^{3} A b \,d^{2} e +\frac {1}{3} x^{3} A c \,d^{3}+x^{3} B a \,d^{2} e +\frac {1}{3} B b \,d^{3} x^{3}+\frac {3}{2} x^{2} A a \,d^{2} e +\frac {1}{2} A b \,d^{3} x^{2}+\frac {1}{2} x^{2} B a \,d^{3}+A \,d^{3} a x\) | \(248\) |
| parallelrisch | \(\frac {1}{7} B \,e^{3} c \,x^{7}+\frac {1}{6} x^{6} A c \,e^{3}+\frac {1}{6} x^{6} B \,e^{3} b +\frac {1}{2} x^{6} B c d \,e^{2}+\frac {1}{5} x^{5} A b \,e^{3}+\frac {3}{5} x^{5} A c d \,e^{2}+\frac {1}{5} x^{5} B \,e^{3} a +\frac {3}{5} x^{5} B b d \,e^{2}+\frac {3}{5} x^{5} B c \,d^{2} e +\frac {1}{4} x^{4} A a \,e^{3}+\frac {3}{4} x^{4} A b d \,e^{2}+\frac {3}{4} x^{4} A c \,d^{2} e +\frac {3}{4} x^{4} B a d \,e^{2}+\frac {3}{4} x^{4} B b \,d^{2} e +\frac {1}{4} x^{4} B c \,d^{3}+x^{3} A a d \,e^{2}+x^{3} A b \,d^{2} e +\frac {1}{3} x^{3} A c \,d^{3}+x^{3} B a \,d^{2} e +\frac {1}{3} B b \,d^{3} x^{3}+\frac {3}{2} x^{2} A a \,d^{2} e +\frac {1}{2} A b \,d^{3} x^{2}+\frac {1}{2} x^{2} B a \,d^{3}+A \,d^{3} a x\) | \(248\) |
| orering | \(\frac {x \left (60 B \,e^{3} c \,x^{6}+70 A c \,e^{3} x^{5}+70 B \,e^{3} b \,x^{5}+210 B c d \,e^{2} x^{5}+84 x^{4} A b \,e^{3}+252 A c d \,e^{2} x^{4}+84 x^{4} B \,e^{3} a +252 x^{4} B b d \,e^{2}+252 B c \,d^{2} e \,x^{4}+105 x^{3} A a \,e^{3}+315 x^{3} A b d \,e^{2}+315 A c \,d^{2} e \,x^{3}+315 x^{3} B a d \,e^{2}+315 x^{3} B b \,d^{2} e +105 B c \,d^{3} x^{3}+420 x^{2} A a d \,e^{2}+420 x^{2} A b \,d^{2} e +140 A c \,d^{3} x^{2}+420 x^{2} B a \,d^{2} e +140 B b \,d^{3} x^{2}+630 x A a \,d^{2} e +210 A b \,d^{3} x +210 B a \,d^{3} x +420 A \,d^{3} a \right )}{420}\) | \(248\) |
Input:
int((B*x+A)*(e*x+d)^3*(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
1/7*B*e^3*c*x^7+(1/6*A*c*e^3+1/6*B*e^3*b+1/2*B*c*d*e^2)*x^6+(1/5*A*b*e^3+3 /5*A*c*d*e^2+1/5*B*e^3*a+3/5*B*b*d*e^2+3/5*B*c*d^2*e)*x^5+(1/4*A*a*e^3+3/4 *A*b*d*e^2+3/4*A*c*d^2*e+3/4*B*a*d*e^2+3/4*B*b*d^2*e+1/4*B*c*d^3)*x^4+(A*a *d*e^2+A*b*d^2*e+1/3*A*c*d^3+B*a*d^2*e+1/3*B*b*d^3)*x^3+(3/2*A*a*d^2*e+1/2 *A*b*d^3+1/2*B*a*d^3)*x^2+A*d^3*a*x
Time = 0.08 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.41 \[ \int (A+B x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {1}{7} \, B c e^{3} x^{7} + \frac {1}{6} \, {\left (3 \, B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{6} + A a d^{3} x + \frac {1}{5} \, {\left (3 \, B c d^{2} e + 3 \, {\left (B b + A c\right )} d e^{2} + {\left (B a + A b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{3} + A a e^{3} + 3 \, {\left (B b + A c\right )} d^{2} e + 3 \, {\left (B a + A b\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a d e^{2} + {\left (B b + A c\right )} d^{3} + 3 \, {\left (B a + A b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a d^{2} e + {\left (B a + A b\right )} d^{3}\right )} x^{2} \] Input:
integrate((B*x+A)*(e*x+d)^3*(c*x^2+b*x+a),x, algorithm="fricas")
Output:
1/7*B*c*e^3*x^7 + 1/6*(3*B*c*d*e^2 + (B*b + A*c)*e^3)*x^6 + A*a*d^3*x + 1/ 5*(3*B*c*d^2*e + 3*(B*b + A*c)*d*e^2 + (B*a + A*b)*e^3)*x^5 + 1/4*(B*c*d^3 + A*a*e^3 + 3*(B*b + A*c)*d^2*e + 3*(B*a + A*b)*d*e^2)*x^4 + 1/3*(3*A*a*d *e^2 + (B*b + A*c)*d^3 + 3*(B*a + A*b)*d^2*e)*x^3 + 1/2*(3*A*a*d^2*e + (B* a + A*b)*d^3)*x^2
Time = 0.03 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.89 \[ \int (A+B x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=A a d^{3} x + \frac {B c e^{3} x^{7}}{7} + x^{6} \left (\frac {A c e^{3}}{6} + \frac {B b e^{3}}{6} + \frac {B c d e^{2}}{2}\right ) + x^{5} \left (\frac {A b e^{3}}{5} + \frac {3 A c d e^{2}}{5} + \frac {B a e^{3}}{5} + \frac {3 B b d e^{2}}{5} + \frac {3 B c d^{2} e}{5}\right ) + x^{4} \left (\frac {A a e^{3}}{4} + \frac {3 A b d e^{2}}{4} + \frac {3 A c d^{2} e}{4} + \frac {3 B a d e^{2}}{4} + \frac {3 B b d^{2} e}{4} + \frac {B c d^{3}}{4}\right ) + x^{3} \left (A a d e^{2} + A b d^{2} e + \frac {A c d^{3}}{3} + B a d^{2} e + \frac {B b d^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 A a d^{2} e}{2} + \frac {A b d^{3}}{2} + \frac {B a d^{3}}{2}\right ) \] Input:
integrate((B*x+A)*(e*x+d)**3*(c*x**2+b*x+a),x)
Output:
A*a*d**3*x + B*c*e**3*x**7/7 + x**6*(A*c*e**3/6 + B*b*e**3/6 + B*c*d*e**2/ 2) + x**5*(A*b*e**3/5 + 3*A*c*d*e**2/5 + B*a*e**3/5 + 3*B*b*d*e**2/5 + 3*B *c*d**2*e/5) + x**4*(A*a*e**3/4 + 3*A*b*d*e**2/4 + 3*A*c*d**2*e/4 + 3*B*a* d*e**2/4 + 3*B*b*d**2*e/4 + B*c*d**3/4) + x**3*(A*a*d*e**2 + A*b*d**2*e + A*c*d**3/3 + B*a*d**2*e + B*b*d**3/3) + x**2*(3*A*a*d**2*e/2 + A*b*d**3/2 + B*a*d**3/2)
Time = 0.04 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.41 \[ \int (A+B x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {1}{7} \, B c e^{3} x^{7} + \frac {1}{6} \, {\left (3 \, B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{6} + A a d^{3} x + \frac {1}{5} \, {\left (3 \, B c d^{2} e + 3 \, {\left (B b + A c\right )} d e^{2} + {\left (B a + A b\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{3} + A a e^{3} + 3 \, {\left (B b + A c\right )} d^{2} e + 3 \, {\left (B a + A b\right )} d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a d e^{2} + {\left (B b + A c\right )} d^{3} + 3 \, {\left (B a + A b\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a d^{2} e + {\left (B a + A b\right )} d^{3}\right )} x^{2} \] Input:
integrate((B*x+A)*(e*x+d)^3*(c*x^2+b*x+a),x, algorithm="maxima")
Output:
1/7*B*c*e^3*x^7 + 1/6*(3*B*c*d*e^2 + (B*b + A*c)*e^3)*x^6 + A*a*d^3*x + 1/ 5*(3*B*c*d^2*e + 3*(B*b + A*c)*d*e^2 + (B*a + A*b)*e^3)*x^5 + 1/4*(B*c*d^3 + A*a*e^3 + 3*(B*b + A*c)*d^2*e + 3*(B*a + A*b)*d*e^2)*x^4 + 1/3*(3*A*a*d *e^2 + (B*b + A*c)*d^3 + 3*(B*a + A*b)*d^2*e)*x^3 + 1/2*(3*A*a*d^2*e + (B* a + A*b)*d^3)*x^2
Time = 0.14 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.86 \[ \int (A+B x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {1}{7} \, B c e^{3} x^{7} + \frac {1}{2} \, B c d e^{2} x^{6} + \frac {1}{6} \, B b e^{3} x^{6} + \frac {1}{6} \, A c e^{3} x^{6} + \frac {3}{5} \, B c d^{2} e x^{5} + \frac {3}{5} \, B b d e^{2} x^{5} + \frac {3}{5} \, A c d e^{2} x^{5} + \frac {1}{5} \, B a e^{3} x^{5} + \frac {1}{5} \, A b e^{3} x^{5} + \frac {1}{4} \, B c d^{3} x^{4} + \frac {3}{4} \, B b d^{2} e x^{4} + \frac {3}{4} \, A c d^{2} e x^{4} + \frac {3}{4} \, B a d e^{2} x^{4} + \frac {3}{4} \, A b d e^{2} x^{4} + \frac {1}{4} \, A a e^{3} x^{4} + \frac {1}{3} \, B b d^{3} x^{3} + \frac {1}{3} \, A c d^{3} x^{3} + B a d^{2} e x^{3} + A b d^{2} e x^{3} + A a d e^{2} x^{3} + \frac {1}{2} \, B a d^{3} x^{2} + \frac {1}{2} \, A b d^{3} x^{2} + \frac {3}{2} \, A a d^{2} e x^{2} + A a d^{3} x \] Input:
integrate((B*x+A)*(e*x+d)^3*(c*x^2+b*x+a),x, algorithm="giac")
Output:
1/7*B*c*e^3*x^7 + 1/2*B*c*d*e^2*x^6 + 1/6*B*b*e^3*x^6 + 1/6*A*c*e^3*x^6 + 3/5*B*c*d^2*e*x^5 + 3/5*B*b*d*e^2*x^5 + 3/5*A*c*d*e^2*x^5 + 1/5*B*a*e^3*x^ 5 + 1/5*A*b*e^3*x^5 + 1/4*B*c*d^3*x^4 + 3/4*B*b*d^2*e*x^4 + 3/4*A*c*d^2*e* x^4 + 3/4*B*a*d*e^2*x^4 + 3/4*A*b*d*e^2*x^4 + 1/4*A*a*e^3*x^4 + 1/3*B*b*d^ 3*x^3 + 1/3*A*c*d^3*x^3 + B*a*d^2*e*x^3 + A*b*d^2*e*x^3 + A*a*d*e^2*x^3 + 1/2*B*a*d^3*x^2 + 1/2*A*b*d^3*x^2 + 3/2*A*a*d^2*e*x^2 + A*a*d^3*x
Time = 11.98 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.55 \[ \int (A+B x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=x^2\,\left (\frac {A\,b\,d^3}{2}+\frac {B\,a\,d^3}{2}+\frac {3\,A\,a\,d^2\,e}{2}\right )+x^6\,\left (\frac {A\,c\,e^3}{6}+\frac {B\,b\,e^3}{6}+\frac {B\,c\,d\,e^2}{2}\right )+x^3\,\left (\frac {A\,c\,d^3}{3}+\frac {B\,b\,d^3}{3}+A\,a\,d\,e^2+A\,b\,d^2\,e+B\,a\,d^2\,e\right )+x^5\,\left (\frac {A\,b\,e^3}{5}+\frac {B\,a\,e^3}{5}+\frac {3\,A\,c\,d\,e^2}{5}+\frac {3\,B\,b\,d\,e^2}{5}+\frac {3\,B\,c\,d^2\,e}{5}\right )+x^4\,\left (\frac {A\,a\,e^3}{4}+\frac {B\,c\,d^3}{4}+\frac {3\,A\,b\,d\,e^2}{4}+\frac {3\,B\,a\,d\,e^2}{4}+\frac {3\,A\,c\,d^2\,e}{4}+\frac {3\,B\,b\,d^2\,e}{4}\right )+A\,a\,d^3\,x+\frac {B\,c\,e^3\,x^7}{7} \] Input:
int((A + B*x)*(d + e*x)^3*(a + b*x + c*x^2),x)
Output:
x^2*((A*b*d^3)/2 + (B*a*d^3)/2 + (3*A*a*d^2*e)/2) + x^6*((A*c*e^3)/6 + (B* b*e^3)/6 + (B*c*d*e^2)/2) + x^3*((A*c*d^3)/3 + (B*b*d^3)/3 + A*a*d*e^2 + A *b*d^2*e + B*a*d^2*e) + x^5*((A*b*e^3)/5 + (B*a*e^3)/5 + (3*A*c*d*e^2)/5 + (3*B*b*d*e^2)/5 + (3*B*c*d^2*e)/5) + x^4*((A*a*e^3)/4 + (B*c*d^3)/4 + (3* A*b*d*e^2)/4 + (3*B*a*d*e^2)/4 + (3*A*c*d^2*e)/4 + (3*B*b*d^2*e)/4) + A*a* d^3*x + (B*c*e^3*x^7)/7
Time = 0.23 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.62 \[ \int (A+B x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {x \left (60 b c \,e^{3} x^{6}+70 a c \,e^{3} x^{5}+70 b^{2} e^{3} x^{5}+210 b c d \,e^{2} x^{5}+168 a b \,e^{3} x^{4}+252 a c d \,e^{2} x^{4}+252 b^{2} d \,e^{2} x^{4}+252 b c \,d^{2} e \,x^{4}+105 a^{2} e^{3} x^{3}+630 a b d \,e^{2} x^{3}+315 a c \,d^{2} e \,x^{3}+315 b^{2} d^{2} e \,x^{3}+105 b c \,d^{3} x^{3}+420 a^{2} d \,e^{2} x^{2}+840 a b \,d^{2} e \,x^{2}+140 a c \,d^{3} x^{2}+140 b^{2} d^{3} x^{2}+630 a^{2} d^{2} e x +420 a b \,d^{3} x +420 a^{2} d^{3}\right )}{420} \] Input:
int((B*x+A)*(e*x+d)^3*(c*x^2+b*x+a),x)
Output:
(x*(420*a**2*d**3 + 630*a**2*d**2*e*x + 420*a**2*d*e**2*x**2 + 105*a**2*e* *3*x**3 + 420*a*b*d**3*x + 840*a*b*d**2*e*x**2 + 630*a*b*d*e**2*x**3 + 168 *a*b*e**3*x**4 + 140*a*c*d**3*x**2 + 315*a*c*d**2*e*x**3 + 252*a*c*d*e**2* x**4 + 70*a*c*e**3*x**5 + 140*b**2*d**3*x**2 + 315*b**2*d**2*e*x**3 + 252* b**2*d*e**2*x**4 + 70*b**2*e**3*x**5 + 105*b*c*d**3*x**3 + 252*b*c*d**2*e* x**4 + 210*b*c*d*e**2*x**5 + 60*b*c*e**3*x**6))/420