\(\int (c+d x)^3 (a+b x^2) (A+B x+C x^2+D x^3) \, dx\) [1]

Optimal result

Integrand size = 30, antiderivative size = 256 \[ \int (c+d x)^3 \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\left (b c^2+a d^2\right ) \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^4}{4 d^6}-\frac {\left (a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b c \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) (c+d x)^5}{5 d^6}+\frac {\left (a d^2 (C d-3 c D)+b \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) (c+d x)^6}{6 d^6}+\frac {\left (a d^2 D-b \left (4 c C d-B d^2-10 c^2 D\right )\right ) (c+d x)^7}{7 d^6}+\frac {b (C d-5 c D) (c+d x)^8}{8 d^6}+\frac {b D (c+d x)^9}{9 d^6} \] Output:

1/4*(a*d^2+b*c^2)*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^4/d^6-1/5*(a*d^2*( 
-B*d^2+2*C*c*d-3*D*c^2)+b*c*(2*A*d^3-3*B*c*d^2+4*C*c^2*d-5*D*c^3))*(d*x+c) 
^5/d^6+1/6*(a*d^2*(C*d-3*D*c)+b*(A*d^3-3*B*c*d^2+6*C*c^2*d-10*D*c^3))*(d*x 
+c)^6/d^6+1/7*(a*d^2*D-b*(-B*d^2+4*C*c*d-10*D*c^2))*(d*x+c)^7/d^6+1/8*b*(C 
*d-5*D*c)*(d*x+c)^8/d^6+1/9*b*D*(d*x+c)^9/d^6
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.05 \[ \int (c+d x)^3 \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=a A c^3 x+\frac {1}{2} a c^2 (B c+3 A d) x^2+\frac {1}{3} c \left (a c (c C+3 B d)+A \left (b c^2+3 a d^2\right )\right ) x^3+\frac {1}{4} \left (b c^2 (B c+3 A d)+a \left (3 c^2 C d+3 B c d^2+A d^3+c^3 D\right )\right ) x^4+\frac {1}{5} \left (b c \left (c^2 C+3 B c d+3 A d^2\right )+a d \left (3 c C d+B d^2+3 c^2 D\right )\right ) x^5+\frac {1}{6} \left (a d^2 (C d+3 c D)+b \left (3 c^2 C d+3 B c d^2+A d^3+c^3 D\right )\right ) x^6+\frac {1}{7} d \left (a d^2 D+b \left (3 c C d+B d^2+3 c^2 D\right )\right ) x^7+\frac {1}{8} b d^2 (C d+3 c D) x^8+\frac {1}{9} b d^3 D x^9 \] Input:

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

Output:

a*A*c^3*x + (a*c^2*(B*c + 3*A*d)*x^2)/2 + (c*(a*c*(c*C + 3*B*d) + A*(b*c^2 
 + 3*a*d^2))*x^3)/3 + ((b*c^2*(B*c + 3*A*d) + a*(3*c^2*C*d + 3*B*c*d^2 + A 
*d^3 + c^3*D))*x^4)/4 + ((b*c*(c^2*C + 3*B*c*d + 3*A*d^2) + a*d*(3*c*C*d + 
 B*d^2 + 3*c^2*D))*x^5)/5 + ((a*d^2*(C*d + 3*c*D) + b*(3*c^2*C*d + 3*B*c*d 
^2 + A*d^3 + c^3*D))*x^6)/6 + (d*(a*d^2*D + b*(3*c*C*d + B*d^2 + 3*c^2*D)) 
*x^7)/7 + (b*d^2*(C*d + 3*c*D)*x^8)/8 + (b*d^3*D*x^9)/9
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2160, 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.

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

\(\Big \downarrow \) 2160

\(\displaystyle \int \left (\frac {(c+d x)^5 \left (a d^2 (C d-3 c D)+b \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^5}+\frac {(c+d x)^4 \left (-a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )-b c \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^5}+\frac {(c+d x)^3 \left (a d^2+b c^2\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^5}+\frac {(c+d x)^6 \left (a d^2 D-b \left (-B d^2-10 c^2 D+4 c C d\right )\right )}{d^5}+\frac {b (c+d x)^7 (C d-5 c D)}{d^5}+\frac {b D (c+d x)^8}{d^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(c+d x)^6 \left (a d^2 (C d-3 c D)+b \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{6 d^6}-\frac {(c+d x)^5 \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{5 d^6}+\frac {(c+d x)^4 \left (a d^2+b c^2\right ) \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^6}+\frac {(c+d x)^7 \left (a d^2 D-b \left (-B d^2-10 c^2 D+4 c C d\right )\right )}{7 d^6}+\frac {b (c+d x)^8 (C d-5 c D)}{8 d^6}+\frac {b D (c+d x)^9}{9 d^6}\)

Input:

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

Output:

((b*c^2 + a*d^2)*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^4)/(4*d^6) 
- ((a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) + b*c*(4*c^2*C*d - 3*B*c*d^2 + 2*A*d 
^3 - 5*c^3*D))*(c + d*x)^5)/(5*d^6) + ((a*d^2*(C*d - 3*c*D) + b*(6*c^2*C*d 
 - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*(c + d*x)^6)/(6*d^6) + ((a*d^2*D - b*(4* 
c*C*d - B*d^2 - 10*c^2*D))*(c + d*x)^7)/(7*d^6) + (b*(C*d - 5*c*D)*(c + d* 
x)^8)/(8*d^6) + (b*D*(c + d*x)^9)/(9*d^6)
 

Defintions of rubi rules used

rule 2009

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

rule 2160

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] 
:> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, 
 d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
 

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.09

method	result	size
norman	\(\frac {b \,d^{3} D x^{9}}{9}+\left (\frac {1}{8} b \,d^{3} C +\frac {3}{8} b c \,d^{2} D\right ) x^{8}+\left (\frac {1}{7} B b \,d^{3}+\frac {3}{7} b c \,d^{2} C +\frac {1}{7} D a \,d^{3}+\frac {3}{7} D b \,c^{2} d \right ) x^{7}+\left (\frac {1}{6} A b \,d^{3}+\frac {1}{2} B b c \,d^{2}+\frac {1}{6} C a \,d^{3}+\frac {1}{2} C b \,c^{2} d +\frac {1}{2} D a c \,d^{2}+\frac {1}{6} D b \,c^{3}\right ) x^{6}+\left (\frac {3}{5} A b c \,d^{2}+\frac {1}{5} B a \,d^{3}+\frac {3}{5} B b \,c^{2} d +\frac {3}{5} C a c \,d^{2}+\frac {1}{5} b \,c^{3} C +\frac {3}{5} a \,c^{2} d D\right ) x^{5}+\left (\frac {1}{4} A \,d^{3} a +\frac {3}{4} A b \,c^{2} d +\frac {3}{4} a B c \,d^{2}+\frac {1}{4} b B \,c^{3}+\frac {3}{4} a \,c^{2} d C +\frac {1}{4} c^{3} a D\right ) x^{4}+\left (A \,d^{2} a c +\frac {1}{3} A b \,c^{3}+B a \,c^{2} d +\frac {1}{3} C a \,c^{3}\right ) x^{3}+\left (\frac {3}{2} a \,c^{2} d A +\frac {1}{2} c^{3} a B \right ) x^{2}+A a \,c^{3} x\)	\(278\)
default	\(\frac {b \,d^{3} D x^{9}}{9}+\frac {\left (b \,d^{3} C +3 b c \,d^{2} D\right ) x^{8}}{8}+\frac {\left (\left (a \,d^{3}+3 b \,c^{2} d \right ) D+3 b c \,d^{2} C +B b \,d^{3}\right ) x^{7}}{7}+\frac {\left (\left (3 a \,d^{2} c +b \,c^{3}\right ) D+\left (a \,d^{3}+3 b \,c^{2} d \right ) C +3 B b c \,d^{2}+A b \,d^{3}\right ) x^{6}}{6}+\frac {\left (3 a \,c^{2} d D+\left (3 a \,d^{2} c +b \,c^{3}\right ) C +\left (a \,d^{3}+3 b \,c^{2} d \right ) B +3 A b c \,d^{2}\right ) x^{5}}{5}+\frac {\left (c^{3} a D+3 a \,c^{2} d C +\left (3 a \,d^{2} c +b \,c^{3}\right ) B +\left (a \,d^{3}+3 b \,c^{2} d \right ) A \right ) x^{4}}{4}+\frac {\left (C a \,c^{3}+3 B a \,c^{2} d +\left (3 a \,d^{2} c +b \,c^{3}\right ) A \right ) x^{3}}{3}+\frac {\left (3 a \,c^{2} d A +c^{3} a B \right ) x^{2}}{2}+A a \,c^{3} x\)	\(281\)
gosper	\(\frac {1}{4} B b \,c^{3} x^{4}+\frac {3}{5} x^{5} A b c \,d^{2}+\frac {1}{2} x^{6} B b c \,d^{2}+\frac {1}{2} x^{6} C b \,c^{2} d +\frac {1}{2} x^{6} D a c \,d^{2}+\frac {3}{7} x^{7} b c \,d^{2} C +\frac {3}{7} x^{7} D b \,c^{2} d +\frac {3}{8} x^{8} b c \,d^{2} D+\frac {1}{2} B a \,c^{3} x^{2}+\frac {3}{5} x^{5} C a c \,d^{2}+\frac {3}{5} x^{5} B b \,c^{2} d +\frac {1}{3} A b \,c^{3} x^{3}+\frac {3}{4} x^{4} a B c \,d^{2}+\frac {3}{4} x^{4} a \,c^{2} d C +x^{3} A \,d^{2} a c +x^{3} B a \,c^{2} d +\frac {3}{5} x^{5} a \,c^{2} d D+\frac {3}{4} x^{4} A b \,c^{2} d +\frac {3}{2} x^{2} a \,c^{2} d A +\frac {1}{6} x^{6} C a \,d^{3}+\frac {1}{6} x^{6} D b \,c^{3}+A a \,c^{3} x +\frac {1}{3} x^{3} C a \,c^{3}+\frac {1}{4} x^{4} c^{3} a D+\frac {1}{9} b \,d^{3} D x^{9}+\frac {1}{8} x^{8} b \,d^{3} C +\frac {1}{7} x^{7} B b \,d^{3}+\frac {1}{7} x^{7} D a \,d^{3}+\frac {1}{5} x^{5} B a \,d^{3}+\frac {1}{5} x^{5} b \,c^{3} C +\frac {1}{4} x^{4} A \,d^{3} a +\frac {1}{6} x^{6} A b \,d^{3}\)	\(333\)
parallelrisch	\(\frac {1}{4} B b \,c^{3} x^{4}+\frac {3}{5} x^{5} A b c \,d^{2}+\frac {1}{2} x^{6} B b c \,d^{2}+\frac {1}{2} x^{6} C b \,c^{2} d +\frac {1}{2} x^{6} D a c \,d^{2}+\frac {3}{7} x^{7} b c \,d^{2} C +\frac {3}{7} x^{7} D b \,c^{2} d +\frac {3}{8} x^{8} b c \,d^{2} D+\frac {1}{2} B a \,c^{3} x^{2}+\frac {3}{5} x^{5} C a c \,d^{2}+\frac {3}{5} x^{5} B b \,c^{2} d +\frac {1}{3} A b \,c^{3} x^{3}+\frac {3}{4} x^{4} a B c \,d^{2}+\frac {3}{4} x^{4} a \,c^{2} d C +x^{3} A \,d^{2} a c +x^{3} B a \,c^{2} d +\frac {3}{5} x^{5} a \,c^{2} d D+\frac {3}{4} x^{4} A b \,c^{2} d +\frac {3}{2} x^{2} a \,c^{2} d A +\frac {1}{6} x^{6} C a \,d^{3}+\frac {1}{6} x^{6} D b \,c^{3}+A a \,c^{3} x +\frac {1}{3} x^{3} C a \,c^{3}+\frac {1}{4} x^{4} c^{3} a D+\frac {1}{9} b \,d^{3} D x^{9}+\frac {1}{8} x^{8} b \,d^{3} C +\frac {1}{7} x^{7} B b \,d^{3}+\frac {1}{7} x^{7} D a \,d^{3}+\frac {1}{5} x^{5} B a \,d^{3}+\frac {1}{5} x^{5} b \,c^{3} C +\frac {1}{4} x^{4} A \,d^{3} a +\frac {1}{6} x^{6} A b \,d^{3}\)	\(333\)
orering	\(\frac {x \left (280 b \,d^{3} D x^{8}+315 C b \,d^{3} x^{7}+945 D b c \,d^{2} x^{7}+360 B b \,d^{3} x^{6}+1080 C b c \,d^{2} x^{6}+360 D a \,d^{3} x^{6}+1080 D b \,c^{2} d \,x^{6}+420 A b \,d^{3} x^{5}+1260 B b c \,d^{2} x^{5}+420 C a \,d^{3} x^{5}+1260 C b \,c^{2} d \,x^{5}+1260 D a c \,d^{2} x^{5}+420 D b \,c^{3} x^{5}+1512 A b c \,d^{2} x^{4}+504 B a \,d^{3} x^{4}+1512 B b \,c^{2} d \,x^{4}+1512 C a c \,d^{2} x^{4}+504 C b \,c^{3} x^{4}+1512 D a \,c^{2} d \,x^{4}+630 A a \,d^{3} x^{3}+1890 A b \,c^{2} d \,x^{3}+1890 B a c \,d^{2} x^{3}+630 B \,c^{3} x^{3} b +1890 C a \,c^{2} d \,x^{3}+630 D a \,c^{3} x^{3}+2520 A a c \,d^{2} x^{2}+840 A b \,c^{3} x^{2}+2520 B a \,c^{2} d \,x^{2}+840 C a \,c^{3} x^{2}+3780 A a \,c^{2} d x +1260 B a \,c^{3} x +2520 A a \,c^{3}\right )}{2520}\)	\(334\)

Input:

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

Output:

1/9*b*d^3*D*x^9+(1/8*b*d^3*C+3/8*b*c*d^2*D)*x^8+(1/7*B*b*d^3+3/7*b*c*d^2*C 
+1/7*D*a*d^3+3/7*D*b*c^2*d)*x^7+(1/6*A*b*d^3+1/2*B*b*c*d^2+1/6*C*a*d^3+1/2 
*C*b*c^2*d+1/2*D*a*c*d^2+1/6*D*b*c^3)*x^6+(3/5*A*b*c*d^2+1/5*B*a*d^3+3/5*B 
*b*c^2*d+3/5*C*a*c*d^2+1/5*b*c^3*C+3/5*a*c^2*d*D)*x^5+(1/4*A*d^3*a+3/4*A*b 
*c^2*d+3/4*a*B*c*d^2+1/4*b*B*c^3+3/4*a*c^2*d*C+1/4*c^3*a*D)*x^4+(A*d^2*a*c 
+1/3*A*b*c^3+B*a*c^2*d+1/3*C*a*c^3)*x^3+(3/2*a*c^2*d*A+1/2*c^3*a*B)*x^2+A* 
a*c^3*x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00 \[ \int (c+d x)^3 \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{9} \, D b d^{3} x^{9} + \frac {1}{8} \, {\left (3 \, D b c d^{2} + C b d^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, D b c^{2} d + 3 \, C b c d^{2} + {\left (D a + B b\right )} d^{3}\right )} x^{7} + \frac {1}{6} \, {\left (D b c^{3} + 3 \, C b c^{2} d + 3 \, {\left (D a + B b\right )} c d^{2} + {\left (C a + A b\right )} d^{3}\right )} x^{6} + A a c^{3} x + \frac {1}{5} \, {\left (C b c^{3} + B a d^{3} + 3 \, {\left (D a + B b\right )} c^{2} d + 3 \, {\left (C a + A b\right )} c d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a c d^{2} + A a d^{3} + {\left (D a + B b\right )} c^{3} + 3 \, {\left (C a + A b\right )} c^{2} d\right )} x^{4} + \frac {1}{3} \, {\left (3 \, B a c^{2} d + 3 \, A a c d^{2} + {\left (C a + A b\right )} c^{3}\right )} x^{3} + \frac {1}{2} \, {\left (B a c^{3} + 3 \, A a c^{2} d\right )} x^{2} \] Input:

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

Output:

1/9*D*b*d^3*x^9 + 1/8*(3*D*b*c*d^2 + C*b*d^3)*x^8 + 1/7*(3*D*b*c^2*d + 3*C 
*b*c*d^2 + (D*a + B*b)*d^3)*x^7 + 1/6*(D*b*c^3 + 3*C*b*c^2*d + 3*(D*a + B* 
b)*c*d^2 + (C*a + A*b)*d^3)*x^6 + A*a*c^3*x + 1/5*(C*b*c^3 + B*a*d^3 + 3*( 
D*a + B*b)*c^2*d + 3*(C*a + A*b)*c*d^2)*x^5 + 1/4*(3*B*a*c*d^2 + A*a*d^3 + 
 (D*a + B*b)*c^3 + 3*(C*a + A*b)*c^2*d)*x^4 + 1/3*(3*B*a*c^2*d + 3*A*a*c*d 
^2 + (C*a + A*b)*c^3)*x^3 + 1/2*(B*a*c^3 + 3*A*a*c^2*d)*x^2
 

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.33 \[ \int (c+d x)^3 \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=A a c^{3} x + \frac {D b d^{3} x^{9}}{9} + x^{8} \left (\frac {C b d^{3}}{8} + \frac {3 D b c d^{2}}{8}\right ) + x^{7} \left (\frac {B b d^{3}}{7} + \frac {3 C b c d^{2}}{7} + \frac {D a d^{3}}{7} + \frac {3 D b c^{2} d}{7}\right ) + x^{6} \left (\frac {A b d^{3}}{6} + \frac {B b c d^{2}}{2} + \frac {C a d^{3}}{6} + \frac {C b c^{2} d}{2} + \frac {D a c d^{2}}{2} + \frac {D b c^{3}}{6}\right ) + x^{5} \cdot \left (\frac {3 A b c d^{2}}{5} + \frac {B a d^{3}}{5} + \frac {3 B b c^{2} d}{5} + \frac {3 C a c d^{2}}{5} + \frac {C b c^{3}}{5} + \frac {3 D a c^{2} d}{5}\right ) + x^{4} \left (\frac {A a d^{3}}{4} + \frac {3 A b c^{2} d}{4} + \frac {3 B a c d^{2}}{4} + \frac {B b c^{3}}{4} + \frac {3 C a c^{2} d}{4} + \frac {D a c^{3}}{4}\right ) + x^{3} \left (A a c d^{2} + \frac {A b c^{3}}{3} + B a c^{2} d + \frac {C a c^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 A a c^{2} d}{2} + \frac {B a c^{3}}{2}\right ) \] Input:

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

Output:

A*a*c**3*x + D*b*d**3*x**9/9 + x**8*(C*b*d**3/8 + 3*D*b*c*d**2/8) + x**7*( 
B*b*d**3/7 + 3*C*b*c*d**2/7 + D*a*d**3/7 + 3*D*b*c**2*d/7) + x**6*(A*b*d** 
3/6 + B*b*c*d**2/2 + C*a*d**3/6 + C*b*c**2*d/2 + D*a*c*d**2/2 + D*b*c**3/6 
) + x**5*(3*A*b*c*d**2/5 + B*a*d**3/5 + 3*B*b*c**2*d/5 + 3*C*a*c*d**2/5 + 
C*b*c**3/5 + 3*D*a*c**2*d/5) + x**4*(A*a*d**3/4 + 3*A*b*c**2*d/4 + 3*B*a*c 
*d**2/4 + B*b*c**3/4 + 3*C*a*c**2*d/4 + D*a*c**3/4) + x**3*(A*a*c*d**2 + A 
*b*c**3/3 + B*a*c**2*d + C*a*c**3/3) + x**2*(3*A*a*c**2*d/2 + B*a*c**3/2)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00 \[ \int (c+d x)^3 \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{9} \, D b d^{3} x^{9} + \frac {1}{8} \, {\left (3 \, D b c d^{2} + C b d^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, D b c^{2} d + 3 \, C b c d^{2} + {\left (D a + B b\right )} d^{3}\right )} x^{7} + \frac {1}{6} \, {\left (D b c^{3} + 3 \, C b c^{2} d + 3 \, {\left (D a + B b\right )} c d^{2} + {\left (C a + A b\right )} d^{3}\right )} x^{6} + A a c^{3} x + \frac {1}{5} \, {\left (C b c^{3} + B a d^{3} + 3 \, {\left (D a + B b\right )} c^{2} d + 3 \, {\left (C a + A b\right )} c d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a c d^{2} + A a d^{3} + {\left (D a + B b\right )} c^{3} + 3 \, {\left (C a + A b\right )} c^{2} d\right )} x^{4} + \frac {1}{3} \, {\left (3 \, B a c^{2} d + 3 \, A a c d^{2} + {\left (C a + A b\right )} c^{3}\right )} x^{3} + \frac {1}{2} \, {\left (B a c^{3} + 3 \, A a c^{2} d\right )} x^{2} \] Input:

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

Output:

1/9*D*b*d^3*x^9 + 1/8*(3*D*b*c*d^2 + C*b*d^3)*x^8 + 1/7*(3*D*b*c^2*d + 3*C 
*b*c*d^2 + (D*a + B*b)*d^3)*x^7 + 1/6*(D*b*c^3 + 3*C*b*c^2*d + 3*(D*a + B* 
b)*c*d^2 + (C*a + A*b)*d^3)*x^6 + A*a*c^3*x + 1/5*(C*b*c^3 + B*a*d^3 + 3*( 
D*a + B*b)*c^2*d + 3*(C*a + A*b)*c*d^2)*x^5 + 1/4*(3*B*a*c*d^2 + A*a*d^3 + 
 (D*a + B*b)*c^3 + 3*(C*a + A*b)*c^2*d)*x^4 + 1/3*(3*B*a*c^2*d + 3*A*a*c*d 
^2 + (C*a + A*b)*c^3)*x^3 + 1/2*(B*a*c^3 + 3*A*a*c^2*d)*x^2
 

Giac [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.30 \[ \int (c+d x)^3 \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{9} \, D b d^{3} x^{9} + \frac {3}{8} \, D b c d^{2} x^{8} + \frac {1}{8} \, C b d^{3} x^{8} + \frac {3}{7} \, D b c^{2} d x^{7} + \frac {3}{7} \, C b c d^{2} x^{7} + \frac {1}{7} \, D a d^{3} x^{7} + \frac {1}{7} \, B b d^{3} x^{7} + \frac {1}{6} \, D b c^{3} x^{6} + \frac {1}{2} \, C b c^{2} d x^{6} + \frac {1}{2} \, D a c d^{2} x^{6} + \frac {1}{2} \, B b c d^{2} x^{6} + \frac {1}{6} \, C a d^{3} x^{6} + \frac {1}{6} \, A b d^{3} x^{6} + \frac {1}{5} \, C b c^{3} x^{5} + \frac {3}{5} \, D a c^{2} d x^{5} + \frac {3}{5} \, B b c^{2} d x^{5} + \frac {3}{5} \, C a c d^{2} x^{5} + \frac {3}{5} \, A b c d^{2} x^{5} + \frac {1}{5} \, B a d^{3} x^{5} + \frac {1}{4} \, D a c^{3} x^{4} + \frac {1}{4} \, B b c^{3} x^{4} + \frac {3}{4} \, C a c^{2} d x^{4} + \frac {3}{4} \, A b c^{2} d x^{4} + \frac {3}{4} \, B a c d^{2} x^{4} + \frac {1}{4} \, A a d^{3} x^{4} + \frac {1}{3} \, C a c^{3} x^{3} + \frac {1}{3} \, A b c^{3} x^{3} + B a c^{2} d x^{3} + A a c d^{2} x^{3} + \frac {1}{2} \, B a c^{3} x^{2} + \frac {3}{2} \, A a c^{2} d x^{2} + A a c^{3} x \] Input:

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

Output:

1/9*D*b*d^3*x^9 + 3/8*D*b*c*d^2*x^8 + 1/8*C*b*d^3*x^8 + 3/7*D*b*c^2*d*x^7 
+ 3/7*C*b*c*d^2*x^7 + 1/7*D*a*d^3*x^7 + 1/7*B*b*d^3*x^7 + 1/6*D*b*c^3*x^6 
+ 1/2*C*b*c^2*d*x^6 + 1/2*D*a*c*d^2*x^6 + 1/2*B*b*c*d^2*x^6 + 1/6*C*a*d^3* 
x^6 + 1/6*A*b*d^3*x^6 + 1/5*C*b*c^3*x^5 + 3/5*D*a*c^2*d*x^5 + 3/5*B*b*c^2* 
d*x^5 + 3/5*C*a*c*d^2*x^5 + 3/5*A*b*c*d^2*x^5 + 1/5*B*a*d^3*x^5 + 1/4*D*a* 
c^3*x^4 + 1/4*B*b*c^3*x^4 + 3/4*C*a*c^2*d*x^4 + 3/4*A*b*c^2*d*x^4 + 3/4*B* 
a*c*d^2*x^4 + 1/4*A*a*d^3*x^4 + 1/3*C*a*c^3*x^3 + 1/3*A*b*c^3*x^3 + B*a*c^ 
2*d*x^3 + A*a*c*d^2*x^3 + 1/2*B*a*c^3*x^2 + 3/2*A*a*c^2*d*x^2 + A*a*c^3*x
 

Mupad [B] (verification not implemented)

Time = 19.48 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.30 \[ \int (c+d x)^3 \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {a\,c^3\,x^4\,D}{4}+\frac {b\,c^3\,x^6\,D}{6}+\frac {a\,d^3\,x^7\,D}{7}+\frac {b\,d^3\,x^9\,D}{9}+A\,a\,c^3\,x+\frac {B\,a\,c^3\,x^2}{2}+\frac {A\,b\,c^3\,x^3}{3}+\frac {A\,a\,d^3\,x^4}{4}+\frac {C\,a\,c^3\,x^3}{3}+\frac {B\,b\,c^3\,x^4}{4}+\frac {B\,a\,d^3\,x^5}{5}+\frac {A\,b\,d^3\,x^6}{6}+\frac {C\,b\,c^3\,x^5}{5}+\frac {C\,a\,d^3\,x^6}{6}+\frac {B\,b\,d^3\,x^7}{7}+\frac {C\,b\,d^3\,x^8}{8}+B\,a\,c^2\,d\,x^3+\frac {3\,A\,b\,c^2\,d\,x^4}{4}+\frac {3\,B\,a\,c\,d^2\,x^4}{4}+\frac {3\,A\,b\,c\,d^2\,x^5}{5}+\frac {3\,C\,a\,c^2\,d\,x^4}{4}+\frac {3\,B\,b\,c^2\,d\,x^5}{5}+\frac {3\,C\,a\,c\,d^2\,x^5}{5}+\frac {B\,b\,c\,d^2\,x^6}{2}+\frac {C\,b\,c^2\,d\,x^6}{2}+\frac {3\,C\,b\,c\,d^2\,x^7}{7}+\frac {3\,a\,c^2\,d\,x^5\,D}{5}+\frac {a\,c\,d^2\,x^6\,D}{2}+\frac {3\,b\,c^2\,d\,x^7\,D}{7}+\frac {3\,b\,c\,d^2\,x^8\,D}{8}+\frac {3\,A\,a\,c^2\,d\,x^2}{2}+A\,a\,c\,d^2\,x^3 \] Input:

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

Output:

(a*c^3*x^4*D)/4 + (b*c^3*x^6*D)/6 + (a*d^3*x^7*D)/7 + (b*d^3*x^9*D)/9 + A* 
a*c^3*x + (B*a*c^3*x^2)/2 + (A*b*c^3*x^3)/3 + (A*a*d^3*x^4)/4 + (C*a*c^3*x 
^3)/3 + (B*b*c^3*x^4)/4 + (B*a*d^3*x^5)/5 + (A*b*d^3*x^6)/6 + (C*b*c^3*x^5 
)/5 + (C*a*d^3*x^6)/6 + (B*b*d^3*x^7)/7 + (C*b*d^3*x^8)/8 + B*a*c^2*d*x^3 
+ (3*A*b*c^2*d*x^4)/4 + (3*B*a*c*d^2*x^4)/4 + (3*A*b*c*d^2*x^5)/5 + (3*C*a 
*c^2*d*x^4)/4 + (3*B*b*c^2*d*x^5)/5 + (3*C*a*c*d^2*x^5)/5 + (B*b*c*d^2*x^6 
)/2 + (C*b*c^2*d*x^6)/2 + (3*C*b*c*d^2*x^7)/7 + (3*a*c^2*d*x^5*D)/5 + (a*c 
*d^2*x^6*D)/2 + (3*b*c^2*d*x^7*D)/7 + (3*b*c*d^2*x^8*D)/8 + (3*A*a*c^2*d*x 
^2)/2 + A*a*c*d^2*x^3
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.07 \[ \int (c+d x)^3 \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {x \left (140 b \,d^{4} x^{8}+630 b c \,d^{3} x^{7}+180 a \,d^{4} x^{6}+180 b^{2} d^{3} x^{6}+1080 b \,c^{2} d^{2} x^{6}+210 a b \,d^{3} x^{5}+840 a c \,d^{3} x^{5}+630 b^{2} c \,d^{2} x^{5}+840 b \,c^{3} d \,x^{5}+756 a b c \,d^{2} x^{4}+252 a b \,d^{3} x^{4}+1512 a \,c^{2} d^{2} x^{4}+756 b^{2} c^{2} d \,x^{4}+252 b \,c^{4} x^{4}+315 a^{2} d^{3} x^{3}+945 a b \,c^{2} d \,x^{3}+945 a b c \,d^{2} x^{3}+1260 a \,c^{3} d \,x^{3}+315 b^{2} c^{3} x^{3}+1260 a^{2} c \,d^{2} x^{2}+420 a b \,c^{3} x^{2}+1260 a b \,c^{2} d \,x^{2}+420 a \,c^{4} x^{2}+1890 a^{2} c^{2} d x +630 a b \,c^{3} x +1260 a^{2} c^{3}\right )}{1260} \] Input:

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

Output:

(x*(1260*a**2*c**3 + 1890*a**2*c**2*d*x + 1260*a**2*c*d**2*x**2 + 315*a**2 
*d**3*x**3 + 420*a*b*c**3*x**2 + 630*a*b*c**3*x + 945*a*b*c**2*d*x**3 + 12 
60*a*b*c**2*d*x**2 + 756*a*b*c*d**2*x**4 + 945*a*b*c*d**2*x**3 + 210*a*b*d 
**3*x**5 + 252*a*b*d**3*x**4 + 420*a*c**4*x**2 + 1260*a*c**3*d*x**3 + 1512 
*a*c**2*d**2*x**4 + 840*a*c*d**3*x**5 + 180*a*d**4*x**6 + 315*b**2*c**3*x* 
*3 + 756*b**2*c**2*d*x**4 + 630*b**2*c*d**2*x**5 + 180*b**2*d**3*x**6 + 25 
2*b*c**4*x**4 + 840*b*c**3*d*x**5 + 1080*b*c**2*d**2*x**6 + 630*b*c*d**3*x 
**7 + 140*b*d**4*x**8))/1260