\(\int (a+b x) (c+d x)^2 (A+B x+C x^2+D x^3) \, dx\) [15]

Optimal result

Integrand size = 28, antiderivative size = 204 \[ \int (a+b x) (c+d x)^2 \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {(b c-a d) \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^3}{3 d^5}-\frac {\left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )\right ) (c+d x)^4}{4 d^5}+\frac {\left (a d (C d-3 c D)-b \left (3 c C d-B d^2-6 c^2 D\right )\right ) (c+d x)^5}{5 d^5}+\frac {(b C d-4 b c D+a d D) (c+d x)^6}{6 d^5}+\frac {b D (c+d x)^7}{7 d^5} \] Output:

-1/3*(-a*d+b*c)*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^3/d^5-1/4*(a*d*(-B*d 
^2+2*C*c*d-3*D*c^2)-b*(A*d^3-2*B*c*d^2+3*C*c^2*d-4*D*c^3))*(d*x+c)^4/d^5+1 
/5*(a*d*(C*d-3*D*c)-b*(-B*d^2+3*C*c*d-6*D*c^2))*(d*x+c)^5/d^5+1/6*(C*b*d+D 
*a*d-4*D*b*c)*(d*x+c)^6/d^5+1/7*b*D*(d*x+c)^7/d^5
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.90 \[ \int (a+b x) (c+d x)^2 \left (A+B x+C x^2+D x^3\right ) \, dx=a A c^2 x+\frac {1}{2} c (A b c+a B c+2 a A d) x^2+\frac {1}{3} \left (b B c^2+a c^2 C+2 A b c d+2 a B c d+a A d^2\right ) x^3+\frac {1}{4} \left (b c^2 C+2 b B c d+2 a c C d+A b d^2+a B d^2+a c^2 D\right ) x^4+\frac {1}{5} \left (2 b c C d+b B d^2+a C d^2+b c^2 D+2 a c d D\right ) x^5+\frac {1}{6} d (b C d+2 b c D+a d D) x^6+\frac {1}{7} b d^2 D x^7 \] Input:

Integrate[(a + b*x)*(c + d*x)^2*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

a*A*c^2*x + (c*(A*b*c + a*B*c + 2*a*A*d)*x^2)/2 + ((b*B*c^2 + a*c^2*C + 2* 
A*b*c*d + 2*a*B*c*d + a*A*d^2)*x^3)/3 + ((b*c^2*C + 2*b*B*c*d + 2*a*c*C*d 
+ A*b*d^2 + a*B*d^2 + a*c^2*D)*x^4)/4 + ((2*b*c*C*d + b*B*d^2 + a*C*d^2 + 
b*c^2*D + 2*a*c*d*D)*x^5)/5 + (d*(b*C*d + 2*b*c*D + a*d*D)*x^6)/6 + (b*d^2 
*D*x^7)/7
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 204, 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.

\(\displaystyle \int (a+b x) (c+d x)^2 \left (A+B x+C x^2+D x^3\right ) \, dx\)

\(\Big \downarrow \) 2123

\(\displaystyle \int \left (\frac {(c+d x)^3 \left (b \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )-a d \left (-B d^2-3 c^2 D+2 c C d\right )\right )}{d^4}+\frac {(c+d x)^2 (a d-b c) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^4}+\frac {(c+d x)^4 \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{d^4}+\frac {(c+d x)^5 (a d D-4 b c D+b C d)}{d^4}+\frac {b D (c+d x)^6}{d^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(c+d x)^4 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (A d^3-2 B c d^2-4 c^3 D+3 c^2 C d\right )\right )}{4 d^5}-\frac {(c+d x)^3 (b c-a d) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^5}+\frac {(c+d x)^5 \left (a d (C d-3 c D)-b \left (-B d^2-6 c^2 D+3 c C d\right )\right )}{5 d^5}+\frac {(c+d x)^6 (a d D-4 b c D+b C d)}{6 d^5}+\frac {b D (c+d x)^7}{7 d^5}\)

Input:

Int[(a + b*x)*(c + d*x)^2*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

-1/3*((b*c - a*d)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^3)/d^5 - ( 
(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(3*c^2*C*d - 2*B*c*d^2 + A*d^3 - 4*c^ 
3*D))*(c + d*x)^4)/(4*d^5) + ((a*d*(C*d - 3*c*D) - b*(3*c*C*d - B*d^2 - 6* 
c^2*D))*(c + d*x)^5)/(5*d^5) + ((b*C*d - 4*b*c*D + a*d*D)*(c + d*x)^6)/(6* 
d^5) + (b*D*(c + d*x)^7)/(7*d^5)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2123

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])
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {b \,d^{2} D x^{7}}{7}+\frac {\left (\left (a \,d^{2}+2 b c d \right ) D+b \,d^{2} C \right ) x^{6}}{6}+\frac {\left (\left (2 a c d +b \,c^{2}\right ) D+\left (a \,d^{2}+2 b c d \right ) C +b B \,d^{2}\right ) x^{5}}{5}+\frac {\left (a \,c^{2} D+\left (2 a c d +b \,c^{2}\right ) C +\left (a \,d^{2}+2 b c d \right ) B +b \,d^{2} A \right ) x^{4}}{4}+\frac {\left (a \,c^{2} C +\left (2 a c d +b \,c^{2}\right ) B +\left (a \,d^{2}+2 b c d \right ) A \right ) x^{3}}{3}+\frac {\left (a \,c^{2} B +\left (2 a c d +b \,c^{2}\right ) A \right ) x^{2}}{2}+a \,c^{2} A x\)	\(189\)
norman	\(\frac {b \,d^{2} D x^{7}}{7}+\left (\frac {1}{6} b \,d^{2} C +\frac {1}{6} D a \,d^{2}+\frac {1}{3} D b c d \right ) x^{6}+\left (\frac {1}{5} b B \,d^{2}+\frac {1}{5} C a \,d^{2}+\frac {2}{5} C b c d +\frac {2}{5} D a c d +\frac {1}{5} D b \,c^{2}\right ) x^{5}+\left (\frac {1}{4} b \,d^{2} A +\frac {1}{4} B a \,d^{2}+\frac {1}{2} B b c d +\frac {1}{2} C a c d +\frac {1}{4} C b \,c^{2}+\frac {1}{4} a \,c^{2} D\right ) x^{4}+\left (\frac {1}{3} A a \,d^{2}+\frac {2}{3} A b c d +\frac {2}{3} B a c d +\frac {1}{3} B b \,c^{2}+\frac {1}{3} a \,c^{2} C \right ) x^{3}+\left (A a c d +\frac {1}{2} A b \,c^{2}+\frac {1}{2} a \,c^{2} B \right ) x^{2}+a \,c^{2} A x\)	\(189\)
orering	\(\frac {x \left (60 b \,d^{2} D x^{6}+70 C b \,d^{2} x^{5}+70 D a \,d^{2} x^{5}+140 D b c d \,x^{5}+84 B b \,d^{2} x^{4}+84 C a \,d^{2} x^{4}+168 C b c d \,x^{4}+168 D a c d \,x^{4}+84 D b \,c^{2} x^{4}+105 A b \,d^{2} x^{3}+105 B a \,d^{2} x^{3}+210 B b c d \,x^{3}+210 C a c d \,x^{3}+105 C b \,c^{2} x^{3}+105 D a \,c^{2} x^{3}+140 A a \,d^{2} x^{2}+280 A b c d \,x^{2}+280 B a c d \,x^{2}+140 B b \,c^{2} x^{2}+140 C a \,c^{2} x^{2}+420 A a c d x +210 A b \,c^{2} x +210 B a \,c^{2} x +420 a \,c^{2} A \right )}{420}\)	\(228\)
gosper	\(\frac {1}{7} b \,d^{2} D x^{7}+\frac {1}{6} x^{6} b \,d^{2} C +\frac {1}{6} x^{6} D a \,d^{2}+\frac {1}{3} x^{6} D b c d +\frac {1}{5} x^{5} b B \,d^{2}+\frac {1}{5} x^{5} C a \,d^{2}+\frac {2}{5} x^{5} C b c d +\frac {2}{5} x^{5} D a c d +\frac {1}{5} x^{5} D b \,c^{2}+\frac {1}{4} x^{4} b \,d^{2} A +\frac {1}{4} x^{4} B a \,d^{2}+\frac {1}{2} x^{4} B b c d +\frac {1}{2} x^{4} C a c d +\frac {1}{4} x^{4} C b \,c^{2}+\frac {1}{4} x^{4} a \,c^{2} D+\frac {1}{3} x^{3} A a \,d^{2}+\frac {2}{3} x^{3} A b c d +\frac {2}{3} x^{3} B a c d +\frac {1}{3} x^{3} B b \,c^{2}+\frac {1}{3} x^{3} a \,c^{2} C +x^{2} A a c d +\frac {1}{2} x^{2} A b \,c^{2}+\frac {1}{2} x^{2} a \,c^{2} B +a \,c^{2} A x\)	\(230\)
parallelrisch	\(\frac {1}{7} b \,d^{2} D x^{7}+\frac {1}{6} x^{6} b \,d^{2} C +\frac {1}{6} x^{6} D a \,d^{2}+\frac {1}{3} x^{6} D b c d +\frac {1}{5} x^{5} b B \,d^{2}+\frac {1}{5} x^{5} C a \,d^{2}+\frac {2}{5} x^{5} C b c d +\frac {2}{5} x^{5} D a c d +\frac {1}{5} x^{5} D b \,c^{2}+\frac {1}{4} x^{4} b \,d^{2} A +\frac {1}{4} x^{4} B a \,d^{2}+\frac {1}{2} x^{4} B b c d +\frac {1}{2} x^{4} C a c d +\frac {1}{4} x^{4} C b \,c^{2}+\frac {1}{4} x^{4} a \,c^{2} D+\frac {1}{3} x^{3} A a \,d^{2}+\frac {2}{3} x^{3} A b c d +\frac {2}{3} x^{3} B a c d +\frac {1}{3} x^{3} B b \,c^{2}+\frac {1}{3} x^{3} a \,c^{2} C +x^{2} A a c d +\frac {1}{2} x^{2} A b \,c^{2}+\frac {1}{2} x^{2} a \,c^{2} B +a \,c^{2} A x\)	\(230\)

Input:

int((b*x+a)*(d*x+c)^2*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVERBOSE)
 

Output:

1/7*b*d^2*D*x^7+1/6*((a*d^2+2*b*c*d)*D+b*d^2*C)*x^6+1/5*((2*a*c*d+b*c^2)*D 
+(a*d^2+2*b*c*d)*C+b*B*d^2)*x^5+1/4*(a*c^2*D+(2*a*c*d+b*c^2)*C+(a*d^2+2*b* 
c*d)*B+b*d^2*A)*x^4+1/3*(a*c^2*C+(2*a*c*d+b*c^2)*B+(a*d^2+2*b*c*d)*A)*x^3+ 
1/2*(a*c^2*B+(2*a*c*d+b*c^2)*A)*x^2+a*c^2*A*x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.84 \[ \int (a+b x) (c+d x)^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{7} \, D b d^{2} x^{7} + \frac {1}{6} \, {\left (2 \, D b c d + {\left (D a + C b\right )} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (D b c^{2} + 2 \, {\left (D a + C b\right )} c d + {\left (C a + B b\right )} d^{2}\right )} x^{5} + A a c^{2} x + \frac {1}{4} \, {\left ({\left (D a + C b\right )} c^{2} + 2 \, {\left (C a + B b\right )} c d + {\left (B a + A b\right )} d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a d^{2} + {\left (C a + B b\right )} c^{2} + 2 \, {\left (B a + A b\right )} c d\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a c d + {\left (B a + A b\right )} c^{2}\right )} x^{2} \] Input:

integrate((b*x+a)*(d*x+c)^2*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")
 

Output:

1/7*D*b*d^2*x^7 + 1/6*(2*D*b*c*d + (D*a + C*b)*d^2)*x^6 + 1/5*(D*b*c^2 + 2 
*(D*a + C*b)*c*d + (C*a + B*b)*d^2)*x^5 + A*a*c^2*x + 1/4*((D*a + C*b)*c^2 
 + 2*(C*a + B*b)*c*d + (B*a + A*b)*d^2)*x^4 + 1/3*(A*a*d^2 + (C*a + B*b)*c 
^2 + 2*(B*a + A*b)*c*d)*x^3 + 1/2*(2*A*a*c*d + (B*a + A*b)*c^2)*x^2
 

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.12 \[ \int (a+b x) (c+d x)^2 \left (A+B x+C x^2+D x^3\right ) \, dx=A a c^{2} x + \frac {D b d^{2} x^{7}}{7} + x^{6} \left (\frac {C b d^{2}}{6} + \frac {D a d^{2}}{6} + \frac {D b c d}{3}\right ) + x^{5} \left (\frac {B b d^{2}}{5} + \frac {C a d^{2}}{5} + \frac {2 C b c d}{5} + \frac {2 D a c d}{5} + \frac {D b c^{2}}{5}\right ) + x^{4} \left (\frac {A b d^{2}}{4} + \frac {B a d^{2}}{4} + \frac {B b c d}{2} + \frac {C a c d}{2} + \frac {C b c^{2}}{4} + \frac {D a c^{2}}{4}\right ) + x^{3} \left (\frac {A a d^{2}}{3} + \frac {2 A b c d}{3} + \frac {2 B a c d}{3} + \frac {B b c^{2}}{3} + \frac {C a c^{2}}{3}\right ) + x^{2} \left (A a c d + \frac {A b c^{2}}{2} + \frac {B a c^{2}}{2}\right ) \] Input:

integrate((b*x+a)*(d*x+c)**2*(D*x**3+C*x**2+B*x+A),x)
                                                                                    
                                                                                    
 

Output:

A*a*c**2*x + D*b*d**2*x**7/7 + x**6*(C*b*d**2/6 + D*a*d**2/6 + D*b*c*d/3) 
+ x**5*(B*b*d**2/5 + C*a*d**2/5 + 2*C*b*c*d/5 + 2*D*a*c*d/5 + D*b*c**2/5) 
+ x**4*(A*b*d**2/4 + B*a*d**2/4 + B*b*c*d/2 + C*a*c*d/2 + C*b*c**2/4 + D*a 
*c**2/4) + x**3*(A*a*d**2/3 + 2*A*b*c*d/3 + 2*B*a*c*d/3 + B*b*c**2/3 + C*a 
*c**2/3) + x**2*(A*a*c*d + A*b*c**2/2 + B*a*c**2/2)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.84 \[ \int (a+b x) (c+d x)^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{7} \, D b d^{2} x^{7} + \frac {1}{6} \, {\left (2 \, D b c d + {\left (D a + C b\right )} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (D b c^{2} + 2 \, {\left (D a + C b\right )} c d + {\left (C a + B b\right )} d^{2}\right )} x^{5} + A a c^{2} x + \frac {1}{4} \, {\left ({\left (D a + C b\right )} c^{2} + 2 \, {\left (C a + B b\right )} c d + {\left (B a + A b\right )} d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a d^{2} + {\left (C a + B b\right )} c^{2} + 2 \, {\left (B a + A b\right )} c d\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a c d + {\left (B a + A b\right )} c^{2}\right )} x^{2} \] Input:

integrate((b*x+a)*(d*x+c)^2*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")
 

Output:

1/7*D*b*d^2*x^7 + 1/6*(2*D*b*c*d + (D*a + C*b)*d^2)*x^6 + 1/5*(D*b*c^2 + 2 
*(D*a + C*b)*c*d + (C*a + B*b)*d^2)*x^5 + A*a*c^2*x + 1/4*((D*a + C*b)*c^2 
 + 2*(C*a + B*b)*c*d + (B*a + A*b)*d^2)*x^4 + 1/3*(A*a*d^2 + (C*a + B*b)*c 
^2 + 2*(B*a + A*b)*c*d)*x^3 + 1/2*(2*A*a*c*d + (B*a + A*b)*c^2)*x^2
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.12 \[ \int (a+b x) (c+d x)^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{7} \, D b d^{2} x^{7} + \frac {1}{3} \, D b c d x^{6} + \frac {1}{6} \, D a d^{2} x^{6} + \frac {1}{6} \, C b d^{2} x^{6} + \frac {1}{5} \, D b c^{2} x^{5} + \frac {2}{5} \, D a c d x^{5} + \frac {2}{5} \, C b c d x^{5} + \frac {1}{5} \, C a d^{2} x^{5} + \frac {1}{5} \, B b d^{2} x^{5} + \frac {1}{4} \, D a c^{2} x^{4} + \frac {1}{4} \, C b c^{2} x^{4} + \frac {1}{2} \, C a c d x^{4} + \frac {1}{2} \, B b c d x^{4} + \frac {1}{4} \, B a d^{2} x^{4} + \frac {1}{4} \, A b d^{2} x^{4} + \frac {1}{3} \, C a c^{2} x^{3} + \frac {1}{3} \, B b c^{2} x^{3} + \frac {2}{3} \, B a c d x^{3} + \frac {2}{3} \, A b c d x^{3} + \frac {1}{3} \, A a d^{2} x^{3} + \frac {1}{2} \, B a c^{2} x^{2} + \frac {1}{2} \, A b c^{2} x^{2} + A a c d x^{2} + A a c^{2} x \] Input:

integrate((b*x+a)*(d*x+c)^2*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")
 

Output:

1/7*D*b*d^2*x^7 + 1/3*D*b*c*d*x^6 + 1/6*D*a*d^2*x^6 + 1/6*C*b*d^2*x^6 + 1/ 
5*D*b*c^2*x^5 + 2/5*D*a*c*d*x^5 + 2/5*C*b*c*d*x^5 + 1/5*C*a*d^2*x^5 + 1/5* 
B*b*d^2*x^5 + 1/4*D*a*c^2*x^4 + 1/4*C*b*c^2*x^4 + 1/2*C*a*c*d*x^4 + 1/2*B* 
b*c*d*x^4 + 1/4*B*a*d^2*x^4 + 1/4*A*b*d^2*x^4 + 1/3*C*a*c^2*x^3 + 1/3*B*b* 
c^2*x^3 + 2/3*B*a*c*d*x^3 + 2/3*A*b*c*d*x^3 + 1/3*A*a*d^2*x^3 + 1/2*B*a*c^ 
2*x^2 + 1/2*A*b*c^2*x^2 + A*a*c*d*x^2 + A*a*c^2*x
 

Mupad [B] (verification not implemented)

Time = 3.50 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.01 \[ \int (a+b x) (c+d x)^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {a\,x^4\,D\,\left (15\,c^2+24\,c\,d\,x+10\,d^2\,x^2\right )}{60}+\frac {b\,x^5\,D\,\left (21\,c^2+35\,c\,d\,x+15\,d^2\,x^2\right )}{105}+\frac {A\,a\,x\,\left (3\,c^2+3\,c\,d\,x+d^2\,x^2\right )}{3}+\frac {A\,b\,x^2\,\left (6\,c^2+8\,c\,d\,x+3\,d^2\,x^2\right )}{12}+\frac {B\,a\,x^2\,\left (6\,c^2+8\,c\,d\,x+3\,d^2\,x^2\right )}{12}+\frac {B\,b\,x^3\,\left (10\,c^2+15\,c\,d\,x+6\,d^2\,x^2\right )}{30}+\frac {C\,a\,x^3\,\left (10\,c^2+15\,c\,d\,x+6\,d^2\,x^2\right )}{30}+\frac {C\,b\,x^4\,\left (15\,c^2+24\,c\,d\,x+10\,d^2\,x^2\right )}{60} \] Input:

int((a + b*x)*(c + d*x)^2*(A + B*x + C*x^2 + x^3*D),x)
 

Output:

(a*x^4*D*(15*c^2 + 10*d^2*x^2 + 24*c*d*x))/60 + (b*x^5*D*(21*c^2 + 15*d^2* 
x^2 + 35*c*d*x))/105 + (A*a*x*(3*c^2 + d^2*x^2 + 3*c*d*x))/3 + (A*b*x^2*(6 
*c^2 + 3*d^2*x^2 + 8*c*d*x))/12 + (B*a*x^2*(6*c^2 + 3*d^2*x^2 + 8*c*d*x))/ 
12 + (B*b*x^3*(10*c^2 + 6*d^2*x^2 + 15*c*d*x))/30 + (C*a*x^3*(10*c^2 + 6*d 
^2*x^2 + 15*c*d*x))/30 + (C*b*x^4*(15*c^2 + 10*d^2*x^2 + 24*c*d*x))/60
 

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.81 \[ \int (a+b x) (c+d x)^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {x \left (60 b \,d^{3} x^{6}+70 a \,d^{3} x^{5}+210 b c \,d^{2} x^{5}+252 a c \,d^{2} x^{4}+84 b^{2} d^{2} x^{4}+252 b \,c^{2} d \,x^{4}+210 a b \,d^{2} x^{3}+315 a \,c^{2} d \,x^{3}+210 b^{2} c d \,x^{3}+105 b \,c^{3} x^{3}+140 a^{2} d^{2} x^{2}+560 a b c d \,x^{2}+140 a \,c^{3} x^{2}+140 b^{2} c^{2} x^{2}+420 a^{2} c d x +420 a b \,c^{2} x +420 a^{2} c^{2}\right )}{420} \] Input:

int((b*x+a)*(d*x+c)^2*(D*x^3+C*x^2+B*x+A),x)
 

Output:

(x*(420*a**2*c**2 + 420*a**2*c*d*x + 140*a**2*d**2*x**2 + 420*a*b*c**2*x + 
 560*a*b*c*d*x**2 + 210*a*b*d**2*x**3 + 140*a*c**3*x**2 + 315*a*c**2*d*x** 
3 + 252*a*c*d**2*x**4 + 70*a*d**3*x**5 + 140*b**2*c**2*x**2 + 210*b**2*c*d 
*x**3 + 84*b**2*d**2*x**4 + 105*b*c**3*x**3 + 252*b*c**2*d*x**4 + 210*b*c* 
d**2*x**5 + 60*b*d**3*x**6))/420