\(\int (c+d x)^3 (a+b x^2)^2 (A+B x+C x^2+D x^3) \, dx\) [8]

Optimal result

Integrand size = 32, antiderivative size = 455 \[ \int (c+d x)^3 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\left (b c^2+a d^2\right )^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^4}{4 d^8}-\frac {\left (b c^2+a d^2\right ) \left (a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b c \left (6 c^2 C d-5 B c d^2+4 A d^3-7 c^3 D\right )\right ) (c+d x)^5}{5 d^8}+\frac {\left (a^2 d^4 (C d-3 c D)+b^2 c^2 \left (15 c^2 C d-10 B c d^2+6 A d^3-21 c^3 D\right )+2 a b d^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) (c+d x)^6}{6 d^8}+\frac {\left (a^2 d^4 D-2 a b d^2 \left (4 c C d-B d^2-10 c^2 D\right )-b^2 c \left (20 c^2 C d-10 B c d^2+4 A d^3-35 c^3 D\right )\right ) (c+d x)^7}{7 d^8}+\frac {b \left (2 a d^2 (C d-5 c D)+b \left (15 c^2 C d-5 B c d^2+A d^3-35 c^3 D\right )\right ) (c+d x)^8}{8 d^8}+\frac {b \left (2 a d^2 D-b \left (6 c C d-B d^2-21 c^2 D\right )\right ) (c+d x)^9}{9 d^8}+\frac {b^2 (C d-7 c D) (c+d x)^{10}}{10 d^8}+\frac {b^2 D (c+d x)^{11}}{11 d^8} \] Output:

1/4*(a*d^2+b*c^2)^2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^4/d^8-1/5*(a*d^2 
+b*c^2)*(a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)+b*c*(4*A*d^3-5*B*c*d^2+6*C*c^2*d-7 
*D*c^3))*(d*x+c)^5/d^8+1/6*(a^2*d^4*(C*d-3*D*c)+b^2*c^2*(6*A*d^3-10*B*c*d^ 
2+15*C*c^2*d-21*D*c^3)+2*a*b*d^2*(A*d^3-3*B*c*d^2+6*C*c^2*d-10*D*c^3))*(d* 
x+c)^6/d^8+1/7*(a^2*d^4*D-2*a*b*d^2*(-B*d^2+4*C*c*d-10*D*c^2)-b^2*c*(4*A*d 
^3-10*B*c*d^2+20*C*c^2*d-35*D*c^3))*(d*x+c)^7/d^8+1/8*b*(2*a*d^2*(C*d-5*D* 
c)+b*(A*d^3-5*B*c*d^2+15*C*c^2*d-35*D*c^3))*(d*x+c)^8/d^8+1/9*b*(2*a*d^2*D 
-b*(-B*d^2+6*C*c*d-21*D*c^2))*(d*x+c)^9/d^8+1/10*b^2*(C*d-7*D*c)*(d*x+c)^1 
0/d^8+1/11*b^2*D*(d*x+c)^11/d^8
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 426, normalized size of antiderivative = 0.94 \[ \int (c+d x)^3 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=a^2 A c^3 x+\frac {1}{2} a^2 c^2 (B c+3 A d) x^2+\frac {1}{3} a c \left (a c (c C+3 B d)+A \left (2 b c^2+3 a d^2\right )\right ) x^3+\frac {1}{4} a \left (2 b c^2 (B c+3 A d)+a \left (3 c^2 C d+3 B c d^2+A d^3+c^3 D\right )\right ) x^4+\frac {1}{5} \left (A b c \left (b c^2+6 a d^2\right )+a \left (2 b c^2 (c C+3 B d)+a d \left (3 c C d+B d^2+3 c^2 D\right )\right )\right ) x^5+\frac {1}{6} \left (b^2 c^2 (B c+3 A d)+a^2 d^2 (C d+3 c D)+2 a b \left (3 c^2 C d+3 B c d^2+A d^3+c^3 D\right )\right ) x^6+\frac {1}{7} \left (b^2 c \left (c^2 C+3 B c d+3 A d^2\right )+a^2 d^3 D+2 a b d \left (3 c C d+B d^2+3 c^2 D\right )\right ) x^7+\frac {1}{8} b \left (2 a d^2 (C d+3 c D)+b \left (3 c^2 C d+3 B c d^2+A d^3+c^3 D\right )\right ) x^8+\frac {1}{9} b d \left (2 a d^2 D+b \left (3 c C d+B d^2+3 c^2 D\right )\right ) x^9+\frac {1}{10} b^2 d^2 (C d+3 c D) x^{10}+\frac {1}{11} b^2 d^3 D x^{11} \] Input:

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

Output:

a^2*A*c^3*x + (a^2*c^2*(B*c + 3*A*d)*x^2)/2 + (a*c*(a*c*(c*C + 3*B*d) + A* 
(2*b*c^2 + 3*a*d^2))*x^3)/3 + (a*(2*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 + ((A*b*c*(b*c^2 + 6*a*d^2) + a*(2*b*c^2 
*(c*C + 3*B*d) + a*d*(3*c*C*d + B*d^2 + 3*c^2*D)))*x^5)/5 + ((b^2*c^2*(B*c 
 + 3*A*d) + a^2*d^2*(C*d + 3*c*D) + 2*a*b*(3*c^2*C*d + 3*B*c*d^2 + A*d^3 + 
 c^3*D))*x^6)/6 + ((b^2*c*(c^2*C + 3*B*c*d + 3*A*d^2) + a^2*d^3*D + 2*a*b* 
d*(3*c*C*d + B*d^2 + 3*c^2*D))*x^7)/7 + (b*(2*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^8)/8 + (b*d*(2*a*d^2*D + b*(3*c*C*d 
 + B*d^2 + 3*c^2*D))*x^9)/9 + (b^2*d^2*(C*d + 3*c*D)*x^10)/10 + (b^2*d^3*D 
*x^11)/11
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2017, 2341, 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.

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

\(\Big \downarrow \) 2017

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

\(\Big \downarrow \) 2341

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{7} x^7 \left (a^2 d^3 D+2 a b d \left (B d^2+3 c^2 D+3 c C d\right )+b^2 c \left (3 A d^2+3 B c d+c^2 C\right )\right )+\frac {1}{4} a^2 x^4 \left (A d^3+3 B c d^2+c^3 D+3 c^2 C d\right )+a^2 A c^3 x+\frac {1}{5} x^5 \left (A b c \left (6 a d^2+b c^2\right )+a \left (a d \left (B d^2+3 c^2 D+3 c C d\right )+2 b c^2 (3 B d+c C)\right )\right )+\frac {1}{3} a c x^3 \left (A \left (3 a d^2+2 b c^2\right )+a c (3 B d+c C)\right )+\frac {c^2 \left (a+b x^2\right )^3 (3 A d+B c)}{6 b}+\frac {1}{8} b x^8 \left (2 a d^2 (3 c D+C d)+b \left (A d^3+3 B c d^2+c^3 D+3 c^2 C d\right )\right )+\frac {1}{6} a x^6 \left (a d^2 (3 c D+C d)+2 b \left (A d^3+3 B c d^2+c^3 D+3 c^2 C d\right )\right )+\frac {1}{9} b d x^9 \left (2 a d^2 D+b \left (B d^2+3 c^2 D+3 c C d\right )\right )+\frac {1}{10} b^2 d^2 x^{10} (3 c D+C d)+\frac {1}{11} b^2 d^3 D x^{11}\)

Input:

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

Output:

a^2*A*c^3*x + (a*c*(a*c*(c*C + 3*B*d) + A*(2*b*c^2 + 3*a*d^2))*x^3)/3 + (a 
^2*(3*c^2*C*d + 3*B*c*d^2 + A*d^3 + c^3*D)*x^4)/4 + ((A*b*c*(b*c^2 + 6*a*d 
^2) + a*(2*b*c^2*(c*C + 3*B*d) + a*d*(3*c*C*d + B*d^2 + 3*c^2*D)))*x^5)/5 
+ (a*(a*d^2*(C*d + 3*c*D) + 2*b*(3*c^2*C*d + 3*B*c*d^2 + A*d^3 + c^3*D))*x 
^6)/6 + ((b^2*c*(c^2*C + 3*B*c*d + 3*A*d^2) + a^2*d^3*D + 2*a*b*d*(3*c*C*d 
 + B*d^2 + 3*c^2*D))*x^7)/7 + (b*(2*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^8)/8 + (b*d*(2*a*d^2*D + b*(3*c*C*d + B*d^2 + 
 3*c^2*D))*x^9)/9 + (b^2*d^2*(C*d + 3*c*D)*x^10)/10 + (b^2*d^3*D*x^11)/11 
+ (c^2*(B*c + 3*A*d)*(a + b*x^2)^3)/(6*b)
 

Defintions of rubi rules used

rule 2009

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

rule 2017

Int[(Px_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Coeff[Px, x, n - 
 1]*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] + Int[(Px - Coeff[Px, x, n - 1] 
*x^(n - 1))*(a + b*x^n)^p, x] /; FreeQ[{a, b}, x] && PolyQ[Px, x] && IGtQ[p 
, 1] && IGtQ[n, 1] && NeQ[Coeff[Px, x, n - 1], 0] && NeQ[Px, Coeff[Px, x, n 
 - 1]*x^(n - 1)] &&  !MatchQ[Px, (Qx_.)*((c_) + (d_.)*x^(m_))^(q_) /; FreeQ 
[{c, d}, x] && PolyQ[Qx, x] && IGtQ[q, 1] && IGtQ[m, 1] && NeQ[Coeff[Qx*(a 
+ b*x^n)^p, x, m - 1], 0] && GtQ[m*q, n*p]]
 

rule 2341

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

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.07

method	result	size
norman	\(\frac {b^{2} d^{3} D x^{11}}{11}+\left (\frac {1}{10} b^{2} d^{3} C +\frac {3}{10} b^{2} c \,d^{2} D\right ) x^{10}+\left (\frac {1}{9} B \,b^{2} d^{3}+\frac {1}{3} b^{2} c \,d^{2} C +\frac {2}{9} D a b \,d^{3}+\frac {1}{3} D b^{2} c^{2} d \right ) x^{9}+\left (\frac {1}{8} A \,b^{2} d^{3}+\frac {3}{8} b^{2} c \,d^{2} B +\frac {1}{4} C a b \,d^{3}+\frac {3}{8} C \,b^{2} c^{2} d +\frac {3}{4} D a b c \,d^{2}+\frac {1}{8} D b^{2} c^{3}\right ) x^{8}+\left (\frac {3}{7} A \,d^{2} b^{2} c +\frac {2}{7} B a b \,d^{3}+\frac {3}{7} B \,b^{2} c^{2} d +\frac {6}{7} C a b c \,d^{2}+\frac {1}{7} C \,b^{2} c^{3}+\frac {1}{7} a^{2} d^{3} D+\frac {6}{7} D a b \,c^{2} d \right ) x^{7}+\left (\frac {1}{3} A a b \,d^{3}+\frac {1}{2} A \,b^{2} c^{2} d +B a b c \,d^{2}+\frac {1}{6} B \,b^{2} c^{3}+\frac {1}{6} C \,a^{2} d^{3}+C a b \,c^{2} d +\frac {1}{2} D a^{2} c \,d^{2}+\frac {1}{3} D a b \,c^{3}\right ) x^{6}+\left (\frac {6}{5} A a b c \,d^{2}+\frac {1}{5} A \,b^{2} c^{3}+\frac {1}{5} a^{2} B \,d^{3}+\frac {6}{5} B a b \,c^{2} d +\frac {3}{5} C \,a^{2} c \,d^{2}+\frac {2}{5} C a b \,c^{3}+\frac {3}{5} a^{2} c^{2} d D\right ) x^{5}+\left (\frac {1}{4} A \,a^{2} d^{3}+\frac {3}{2} A a b \,c^{2} d +\frac {3}{4} B \,a^{2} c \,d^{2}+\frac {1}{2} B a b \,c^{3}+\frac {3}{4} a^{2} c^{2} d C +\frac {1}{4} a^{2} c^{3} D\right ) x^{4}+\left (A \,d^{2} a^{2} c +\frac {2}{3} A a b \,c^{3}+a^{2} c^{2} d B +\frac {1}{3} C \,a^{2} c^{3}\right ) x^{3}+\left (\frac {3}{2} a^{2} c^{2} d A +\frac {1}{2} B \,a^{2} c^{3}\right ) x^{2}+a^{2} A \,c^{3} x\)	\(486\)
default	\(\frac {b^{2} d^{3} D x^{11}}{11}+\frac {\left (b^{2} d^{3} C +3 b^{2} c \,d^{2} D\right ) x^{10}}{10}+\frac {\left (\left (2 a b \,d^{3}+3 b^{2} c^{2} d \right ) D+3 b^{2} c \,d^{2} C +B \,b^{2} d^{3}\right ) x^{9}}{9}+\frac {\left (\left (6 a b c \,d^{2}+c^{3} b^{2}\right ) D+\left (2 a b \,d^{3}+3 b^{2} c^{2} d \right ) C +3 b^{2} c \,d^{2} B +A \,b^{2} d^{3}\right ) x^{8}}{8}+\frac {\left (\left (a^{2} d^{3}+6 a b \,c^{2} d \right ) D+\left (6 a b c \,d^{2}+c^{3} b^{2}\right ) C +\left (2 a b \,d^{3}+3 b^{2} c^{2} d \right ) B +3 A \,d^{2} b^{2} c \right ) x^{7}}{7}+\frac {\left (\left (3 a^{2} c \,d^{2}+2 a b \,c^{3}\right ) D+\left (a^{2} d^{3}+6 a b \,c^{2} d \right ) C +\left (6 a b c \,d^{2}+c^{3} b^{2}\right ) B +\left (2 a b \,d^{3}+3 b^{2} c^{2} d \right ) A \right ) x^{6}}{6}+\frac {\left (3 a^{2} c^{2} d D+\left (3 a^{2} c \,d^{2}+2 a b \,c^{3}\right ) C +\left (a^{2} d^{3}+6 a b \,c^{2} d \right ) B +\left (6 a b c \,d^{2}+c^{3} b^{2}\right ) A \right ) x^{5}}{5}+\frac {\left (a^{2} c^{3} D+3 a^{2} c^{2} d C +\left (3 a^{2} c \,d^{2}+2 a b \,c^{3}\right ) B +\left (a^{2} d^{3}+6 a b \,c^{2} d \right ) A \right ) x^{4}}{4}+\frac {\left (C \,a^{2} c^{3}+3 a^{2} c^{2} d B +\left (3 a^{2} c \,d^{2}+2 a b \,c^{3}\right ) A \right ) x^{3}}{3}+\frac {\left (3 a^{2} c^{2} d A +B \,a^{2} c^{3}\right ) x^{2}}{2}+a^{2} A \,c^{3} x\)	\(501\)
gosper	\(\frac {2}{3} x^{3} A a b \,c^{3}+\frac {3}{5} x^{5} C \,a^{2} c \,d^{2}+x^{6} B a b c \,d^{2}+\frac {3}{8} x^{8} b^{2} c \,d^{2} B +\frac {1}{4} x^{8} C a b \,d^{3}+\frac {3}{8} x^{8} C \,b^{2} c^{2} d +\frac {3}{10} x^{10} b^{2} c \,d^{2} D+\frac {1}{8} x^{8} D b^{2} c^{3}+\frac {1}{7} x^{7} C \,b^{2} c^{3}+\frac {3}{2} x^{2} a^{2} c^{2} d A +\frac {1}{10} x^{10} b^{2} d^{3} C +\frac {1}{9} x^{9} B \,b^{2} d^{3}+\frac {1}{8} x^{8} A \,b^{2} d^{3}+\frac {1}{3} x^{9} b^{2} c \,d^{2} C +x^{3} a^{2} c^{2} d B +\frac {3}{4} x^{8} D a b c \,d^{2}+\frac {1}{3} x^{3} C \,a^{2} c^{3}+\frac {1}{3} x^{6} A a b \,d^{3}+\frac {1}{2} x^{6} A \,b^{2} c^{2} d +\frac {1}{2} x^{6} D a^{2} c \,d^{2}+\frac {1}{3} x^{9} D b^{2} c^{2} d +\frac {2}{9} x^{9} D a b \,d^{3}+\frac {3}{7} x^{7} A \,d^{2} b^{2} c +\frac {2}{7} x^{7} B a b \,d^{3}+\frac {3}{7} x^{7} B \,b^{2} c^{2} d +\frac {2}{5} x^{5} C a b \,c^{3}+\frac {3}{5} x^{5} a^{2} c^{2} d D+\frac {3}{4} x^{4} B \,a^{2} c \,d^{2}+\frac {1}{2} x^{4} B a b \,c^{3}+\frac {3}{4} x^{4} a^{2} c^{2} d C +x^{3} A \,d^{2} a^{2} c +\frac {1}{3} x^{6} D a b \,c^{3}+\frac {1}{11} b^{2} d^{3} D x^{11}+a^{2} A \,c^{3} x +\frac {1}{2} x^{2} B \,a^{2} c^{3}+\frac {6}{7} x^{7} C a b c \,d^{2}+\frac {6}{5} x^{5} A a b c \,d^{2}+\frac {1}{4} x^{4} A \,a^{2} d^{3}+\frac {1}{4} x^{4} a^{2} c^{3} D+\frac {1}{5} A \,b^{2} c^{3} x^{5}+\frac {6}{7} x^{7} D a b \,c^{2} d +x^{6} C a b \,c^{2} d +\frac {1}{7} x^{7} a^{2} d^{3} D+\frac {6}{5} x^{5} B a b \,c^{2} d +\frac {3}{2} x^{4} A a b \,c^{2} d +\frac {1}{5} x^{5} a^{2} B \,d^{3}+\frac {1}{6} x^{6} C \,a^{2} d^{3}+\frac {1}{6} B \,b^{2} c^{3} x^{6}\)	\(579\)
parallelrisch	\(\frac {2}{3} x^{3} A a b \,c^{3}+\frac {3}{5} x^{5} C \,a^{2} c \,d^{2}+x^{6} B a b c \,d^{2}+\frac {3}{8} x^{8} b^{2} c \,d^{2} B +\frac {1}{4} x^{8} C a b \,d^{3}+\frac {3}{8} x^{8} C \,b^{2} c^{2} d +\frac {3}{10} x^{10} b^{2} c \,d^{2} D+\frac {1}{8} x^{8} D b^{2} c^{3}+\frac {1}{7} x^{7} C \,b^{2} c^{3}+\frac {3}{2} x^{2} a^{2} c^{2} d A +\frac {1}{10} x^{10} b^{2} d^{3} C +\frac {1}{9} x^{9} B \,b^{2} d^{3}+\frac {1}{8} x^{8} A \,b^{2} d^{3}+\frac {1}{3} x^{9} b^{2} c \,d^{2} C +x^{3} a^{2} c^{2} d B +\frac {3}{4} x^{8} D a b c \,d^{2}+\frac {1}{3} x^{3} C \,a^{2} c^{3}+\frac {1}{3} x^{6} A a b \,d^{3}+\frac {1}{2} x^{6} A \,b^{2} c^{2} d +\frac {1}{2} x^{6} D a^{2} c \,d^{2}+\frac {1}{3} x^{9} D b^{2} c^{2} d +\frac {2}{9} x^{9} D a b \,d^{3}+\frac {3}{7} x^{7} A \,d^{2} b^{2} c +\frac {2}{7} x^{7} B a b \,d^{3}+\frac {3}{7} x^{7} B \,b^{2} c^{2} d +\frac {2}{5} x^{5} C a b \,c^{3}+\frac {3}{5} x^{5} a^{2} c^{2} d D+\frac {3}{4} x^{4} B \,a^{2} c \,d^{2}+\frac {1}{2} x^{4} B a b \,c^{3}+\frac {3}{4} x^{4} a^{2} c^{2} d C +x^{3} A \,d^{2} a^{2} c +\frac {1}{3} x^{6} D a b \,c^{3}+\frac {1}{11} b^{2} d^{3} D x^{11}+a^{2} A \,c^{3} x +\frac {1}{2} x^{2} B \,a^{2} c^{3}+\frac {6}{7} x^{7} C a b c \,d^{2}+\frac {6}{5} x^{5} A a b c \,d^{2}+\frac {1}{4} x^{4} A \,a^{2} d^{3}+\frac {1}{4} x^{4} a^{2} c^{3} D+\frac {1}{5} A \,b^{2} c^{3} x^{5}+\frac {6}{7} x^{7} D a b \,c^{2} d +x^{6} C a b \,c^{2} d +\frac {1}{7} x^{7} a^{2} d^{3} D+\frac {6}{5} x^{5} B a b \,c^{2} d +\frac {3}{2} x^{4} A a b \,c^{2} d +\frac {1}{5} x^{5} a^{2} B \,d^{3}+\frac {1}{6} x^{6} C \,a^{2} d^{3}+\frac {1}{6} B \,b^{2} c^{3} x^{6}\)	\(579\)
orering	\(\frac {x \left (2520 b^{2} d^{3} D x^{10}+2772 C \,b^{2} d^{3} x^{9}+8316 D b^{2} c \,d^{2} x^{9}+3080 B \,b^{2} d^{3} x^{8}+9240 C \,b^{2} c \,d^{2} x^{8}+6160 D a b \,d^{3} x^{8}+9240 D b^{2} c^{2} d \,x^{8}+3465 A \,b^{2} d^{3} x^{7}+10395 B \,b^{2} c \,d^{2} x^{7}+6930 C a b \,d^{3} x^{7}+10395 C \,b^{2} c^{2} d \,x^{7}+20790 D a b c \,d^{2} x^{7}+3465 D b^{2} c^{3} x^{7}+11880 A \,b^{2} c \,d^{2} x^{6}+7920 B a b \,d^{3} x^{6}+11880 B \,b^{2} c^{2} d \,x^{6}+23760 C a b c \,d^{2} x^{6}+3960 C \,b^{2} c^{3} x^{6}+3960 D a^{2} d^{3} x^{6}+23760 D a b \,c^{2} d \,x^{6}+9240 A a b \,d^{3} x^{5}+13860 A \,b^{2} c^{2} d \,x^{5}+27720 B a b c \,d^{2} x^{5}+4620 B \,b^{2} c^{3} x^{5}+4620 C \,a^{2} d^{3} x^{5}+27720 C a b \,c^{2} d \,x^{5}+13860 D a^{2} c \,d^{2} x^{5}+9240 D a b \,c^{3} x^{5}+33264 A a b c \,d^{2} x^{4}+5544 A \,b^{2} c^{3} x^{4}+5544 B \,a^{2} d^{3} x^{4}+33264 B a b \,c^{2} d \,x^{4}+16632 C \,a^{2} c \,d^{2} x^{4}+11088 C a b \,c^{3} x^{4}+16632 D a^{2} c^{2} d \,x^{4}+6930 A \,a^{2} d^{3} x^{3}+41580 A a b \,c^{2} d \,x^{3}+20790 B \,a^{2} c \,d^{2} x^{3}+13860 B a b \,c^{3} x^{3}+20790 C \,a^{2} c^{2} d \,x^{3}+6930 D a^{2} c^{3} x^{3}+27720 A \,a^{2} c \,d^{2} x^{2}+18480 A a b \,c^{3} x^{2}+27720 B \,a^{2} c^{2} d \,x^{2}+9240 C \,a^{2} c^{3} x^{2}+41580 A \,a^{2} c^{2} d x +13860 B \,a^{2} c^{3} x +27720 a^{2} A \,c^{3}\right )}{27720}\)	\(582\)

Input:

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

Output:

1/11*b^2*d^3*D*x^11+(1/10*b^2*d^3*C+3/10*b^2*c*d^2*D)*x^10+(1/9*B*b^2*d^3+ 
1/3*b^2*c*d^2*C+2/9*D*a*b*d^3+1/3*D*b^2*c^2*d)*x^9+(1/8*A*b^2*d^3+3/8*b^2* 
c*d^2*B+1/4*C*a*b*d^3+3/8*C*b^2*c^2*d+3/4*D*a*b*c*d^2+1/8*D*b^2*c^3)*x^8+( 
3/7*A*d^2*b^2*c+2/7*B*a*b*d^3+3/7*B*b^2*c^2*d+6/7*C*a*b*c*d^2+1/7*C*b^2*c^ 
3+1/7*a^2*d^3*D+6/7*D*a*b*c^2*d)*x^7+(1/3*A*a*b*d^3+1/2*A*b^2*c^2*d+B*a*b* 
c*d^2+1/6*B*b^2*c^3+1/6*C*a^2*d^3+C*a*b*c^2*d+1/2*D*a^2*c*d^2+1/3*D*a*b*c^ 
3)*x^6+(6/5*A*a*b*c*d^2+1/5*A*b^2*c^3+1/5*a^2*B*d^3+6/5*B*a*b*c^2*d+3/5*C* 
a^2*c*d^2+2/5*C*a*b*c^3+3/5*a^2*c^2*d*D)*x^5+(1/4*A*a^2*d^3+3/2*A*a*b*c^2* 
d+3/4*B*a^2*c*d^2+1/2*B*a*b*c^3+3/4*a^2*c^2*d*C+1/4*a^2*c^3*D)*x^4+(A*d^2* 
a^2*c+2/3*A*a*b*c^3+a^2*c^2*d*B+1/3*C*a^2*c^3)*x^3+(3/2*a^2*c^2*d*A+1/2*B* 
a^2*c^3)*x^2+a^2*A*c^3*x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.01 \[ \int (c+d x)^3 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{11} \, D b^{2} d^{3} x^{11} + \frac {1}{10} \, {\left (3 \, D b^{2} c d^{2} + C b^{2} d^{3}\right )} x^{10} + \frac {1}{9} \, {\left (3 \, D b^{2} c^{2} d + 3 \, C b^{2} c d^{2} + {\left (2 \, D a b + B b^{2}\right )} d^{3}\right )} x^{9} + \frac {1}{8} \, {\left (D b^{2} c^{3} + 3 \, C b^{2} c^{2} d + 3 \, {\left (2 \, D a b + B b^{2}\right )} c d^{2} + {\left (2 \, C a b + A b^{2}\right )} d^{3}\right )} x^{8} + \frac {1}{7} \, {\left (C b^{2} c^{3} + 3 \, {\left (2 \, D a b + B b^{2}\right )} c^{2} d + 3 \, {\left (2 \, C a b + A b^{2}\right )} c d^{2} + {\left (D a^{2} + 2 \, B a b\right )} d^{3}\right )} x^{7} + A a^{2} c^{3} x + \frac {1}{6} \, {\left ({\left (2 \, D a b + B b^{2}\right )} c^{3} + 3 \, {\left (2 \, C a b + A b^{2}\right )} c^{2} d + 3 \, {\left (D a^{2} + 2 \, B a b\right )} c d^{2} + {\left (C a^{2} + 2 \, A a b\right )} d^{3}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{2} d^{3} + {\left (2 \, C a b + A b^{2}\right )} c^{3} + 3 \, {\left (D a^{2} + 2 \, B a b\right )} c^{2} d + 3 \, {\left (C a^{2} + 2 \, A a b\right )} c d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{2} + A a^{2} d^{3} + {\left (D a^{2} + 2 \, B a b\right )} c^{3} + 3 \, {\left (C a^{2} + 2 \, A a b\right )} c^{2} d\right )} x^{4} + \frac {1}{3} \, {\left (3 \, B a^{2} c^{2} d + 3 \, A a^{2} c d^{2} + {\left (C a^{2} + 2 \, A a b\right )} c^{3}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} c^{3} + 3 \, A a^{2} c^{2} d\right )} x^{2} \] Input:

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

Output:

1/11*D*b^2*d^3*x^11 + 1/10*(3*D*b^2*c*d^2 + C*b^2*d^3)*x^10 + 1/9*(3*D*b^2 
*c^2*d + 3*C*b^2*c*d^2 + (2*D*a*b + B*b^2)*d^3)*x^9 + 1/8*(D*b^2*c^3 + 3*C 
*b^2*c^2*d + 3*(2*D*a*b + B*b^2)*c*d^2 + (2*C*a*b + A*b^2)*d^3)*x^8 + 1/7* 
(C*b^2*c^3 + 3*(2*D*a*b + B*b^2)*c^2*d + 3*(2*C*a*b + A*b^2)*c*d^2 + (D*a^ 
2 + 2*B*a*b)*d^3)*x^7 + A*a^2*c^3*x + 1/6*((2*D*a*b + B*b^2)*c^3 + 3*(2*C* 
a*b + A*b^2)*c^2*d + 3*(D*a^2 + 2*B*a*b)*c*d^2 + (C*a^2 + 2*A*a*b)*d^3)*x^ 
6 + 1/5*(B*a^2*d^3 + (2*C*a*b + A*b^2)*c^3 + 3*(D*a^2 + 2*B*a*b)*c^2*d + 3 
*(C*a^2 + 2*A*a*b)*c*d^2)*x^5 + 1/4*(3*B*a^2*c*d^2 + A*a^2*d^3 + (D*a^2 + 
2*B*a*b)*c^3 + 3*(C*a^2 + 2*A*a*b)*c^2*d)*x^4 + 1/3*(3*B*a^2*c^2*d + 3*A*a 
^2*c*d^2 + (C*a^2 + 2*A*a*b)*c^3)*x^3 + 1/2*(B*a^2*c^3 + 3*A*a^2*c^2*d)*x^ 
2
 

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.30 \[ \int (c+d x)^3 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=A a^{2} c^{3} x + \frac {D b^{2} d^{3} x^{11}}{11} + x^{10} \left (\frac {C b^{2} d^{3}}{10} + \frac {3 D b^{2} c d^{2}}{10}\right ) + x^{9} \left (\frac {B b^{2} d^{3}}{9} + \frac {C b^{2} c d^{2}}{3} + \frac {2 D a b d^{3}}{9} + \frac {D b^{2} c^{2} d}{3}\right ) + x^{8} \left (\frac {A b^{2} d^{3}}{8} + \frac {3 B b^{2} c d^{2}}{8} + \frac {C a b d^{3}}{4} + \frac {3 C b^{2} c^{2} d}{8} + \frac {3 D a b c d^{2}}{4} + \frac {D b^{2} c^{3}}{8}\right ) + x^{7} \cdot \left (\frac {3 A b^{2} c d^{2}}{7} + \frac {2 B a b d^{3}}{7} + \frac {3 B b^{2} c^{2} d}{7} + \frac {6 C a b c d^{2}}{7} + \frac {C b^{2} c^{3}}{7} + \frac {D a^{2} d^{3}}{7} + \frac {6 D a b c^{2} d}{7}\right ) + x^{6} \left (\frac {A a b d^{3}}{3} + \frac {A b^{2} c^{2} d}{2} + B a b c d^{2} + \frac {B b^{2} c^{3}}{6} + \frac {C a^{2} d^{3}}{6} + C a b c^{2} d + \frac {D a^{2} c d^{2}}{2} + \frac {D a b c^{3}}{3}\right ) + x^{5} \cdot \left (\frac {6 A a b c d^{2}}{5} + \frac {A b^{2} c^{3}}{5} + \frac {B a^{2} d^{3}}{5} + \frac {6 B a b c^{2} d}{5} + \frac {3 C a^{2} c d^{2}}{5} + \frac {2 C a b c^{3}}{5} + \frac {3 D a^{2} c^{2} d}{5}\right ) + x^{4} \left (\frac {A a^{2} d^{3}}{4} + \frac {3 A a b c^{2} d}{2} + \frac {3 B a^{2} c d^{2}}{4} + \frac {B a b c^{3}}{2} + \frac {3 C a^{2} c^{2} d}{4} + \frac {D a^{2} c^{3}}{4}\right ) + x^{3} \left (A a^{2} c d^{2} + \frac {2 A a b c^{3}}{3} + B a^{2} c^{2} d + \frac {C a^{2} c^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 A a^{2} c^{2} d}{2} + \frac {B a^{2} c^{3}}{2}\right ) \] Input:

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

Output:

A*a**2*c**3*x + D*b**2*d**3*x**11/11 + x**10*(C*b**2*d**3/10 + 3*D*b**2*c* 
d**2/10) + x**9*(B*b**2*d**3/9 + C*b**2*c*d**2/3 + 2*D*a*b*d**3/9 + D*b**2 
*c**2*d/3) + x**8*(A*b**2*d**3/8 + 3*B*b**2*c*d**2/8 + C*a*b*d**3/4 + 3*C* 
b**2*c**2*d/8 + 3*D*a*b*c*d**2/4 + D*b**2*c**3/8) + x**7*(3*A*b**2*c*d**2/ 
7 + 2*B*a*b*d**3/7 + 3*B*b**2*c**2*d/7 + 6*C*a*b*c*d**2/7 + C*b**2*c**3/7 
+ D*a**2*d**3/7 + 6*D*a*b*c**2*d/7) + x**6*(A*a*b*d**3/3 + A*b**2*c**2*d/2 
 + B*a*b*c*d**2 + B*b**2*c**3/6 + C*a**2*d**3/6 + C*a*b*c**2*d + D*a**2*c* 
d**2/2 + D*a*b*c**3/3) + x**5*(6*A*a*b*c*d**2/5 + A*b**2*c**3/5 + B*a**2*d 
**3/5 + 6*B*a*b*c**2*d/5 + 3*C*a**2*c*d**2/5 + 2*C*a*b*c**3/5 + 3*D*a**2*c 
**2*d/5) + x**4*(A*a**2*d**3/4 + 3*A*a*b*c**2*d/2 + 3*B*a**2*c*d**2/4 + B* 
a*b*c**3/2 + 3*C*a**2*c**2*d/4 + D*a**2*c**3/4) + x**3*(A*a**2*c*d**2 + 2* 
A*a*b*c**3/3 + B*a**2*c**2*d + C*a**2*c**3/3) + x**2*(3*A*a**2*c**2*d/2 + 
B*a**2*c**3/2)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.01 \[ \int (c+d x)^3 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{11} \, D b^{2} d^{3} x^{11} + \frac {1}{10} \, {\left (3 \, D b^{2} c d^{2} + C b^{2} d^{3}\right )} x^{10} + \frac {1}{9} \, {\left (3 \, D b^{2} c^{2} d + 3 \, C b^{2} c d^{2} + {\left (2 \, D a b + B b^{2}\right )} d^{3}\right )} x^{9} + \frac {1}{8} \, {\left (D b^{2} c^{3} + 3 \, C b^{2} c^{2} d + 3 \, {\left (2 \, D a b + B b^{2}\right )} c d^{2} + {\left (2 \, C a b + A b^{2}\right )} d^{3}\right )} x^{8} + \frac {1}{7} \, {\left (C b^{2} c^{3} + 3 \, {\left (2 \, D a b + B b^{2}\right )} c^{2} d + 3 \, {\left (2 \, C a b + A b^{2}\right )} c d^{2} + {\left (D a^{2} + 2 \, B a b\right )} d^{3}\right )} x^{7} + A a^{2} c^{3} x + \frac {1}{6} \, {\left ({\left (2 \, D a b + B b^{2}\right )} c^{3} + 3 \, {\left (2 \, C a b + A b^{2}\right )} c^{2} d + 3 \, {\left (D a^{2} + 2 \, B a b\right )} c d^{2} + {\left (C a^{2} + 2 \, A a b\right )} d^{3}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{2} d^{3} + {\left (2 \, C a b + A b^{2}\right )} c^{3} + 3 \, {\left (D a^{2} + 2 \, B a b\right )} c^{2} d + 3 \, {\left (C a^{2} + 2 \, A a b\right )} c d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{2} + A a^{2} d^{3} + {\left (D a^{2} + 2 \, B a b\right )} c^{3} + 3 \, {\left (C a^{2} + 2 \, A a b\right )} c^{2} d\right )} x^{4} + \frac {1}{3} \, {\left (3 \, B a^{2} c^{2} d + 3 \, A a^{2} c d^{2} + {\left (C a^{2} + 2 \, A a b\right )} c^{3}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} c^{3} + 3 \, A a^{2} c^{2} d\right )} x^{2} \] Input:

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

Output:

1/11*D*b^2*d^3*x^11 + 1/10*(3*D*b^2*c*d^2 + C*b^2*d^3)*x^10 + 1/9*(3*D*b^2 
*c^2*d + 3*C*b^2*c*d^2 + (2*D*a*b + B*b^2)*d^3)*x^9 + 1/8*(D*b^2*c^3 + 3*C 
*b^2*c^2*d + 3*(2*D*a*b + B*b^2)*c*d^2 + (2*C*a*b + A*b^2)*d^3)*x^8 + 1/7* 
(C*b^2*c^3 + 3*(2*D*a*b + B*b^2)*c^2*d + 3*(2*C*a*b + A*b^2)*c*d^2 + (D*a^ 
2 + 2*B*a*b)*d^3)*x^7 + A*a^2*c^3*x + 1/6*((2*D*a*b + B*b^2)*c^3 + 3*(2*C* 
a*b + A*b^2)*c^2*d + 3*(D*a^2 + 2*B*a*b)*c*d^2 + (C*a^2 + 2*A*a*b)*d^3)*x^ 
6 + 1/5*(B*a^2*d^3 + (2*C*a*b + A*b^2)*c^3 + 3*(D*a^2 + 2*B*a*b)*c^2*d + 3 
*(C*a^2 + 2*A*a*b)*c*d^2)*x^5 + 1/4*(3*B*a^2*c*d^2 + A*a^2*d^3 + (D*a^2 + 
2*B*a*b)*c^3 + 3*(C*a^2 + 2*A*a*b)*c^2*d)*x^4 + 1/3*(3*B*a^2*c^2*d + 3*A*a 
^2*c*d^2 + (C*a^2 + 2*A*a*b)*c^3)*x^3 + 1/2*(B*a^2*c^3 + 3*A*a^2*c^2*d)*x^ 
2
 

Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.27 \[ \int (c+d x)^3 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{11} \, D b^{2} d^{3} x^{11} + \frac {3}{10} \, D b^{2} c d^{2} x^{10} + \frac {1}{10} \, C b^{2} d^{3} x^{10} + \frac {1}{3} \, D b^{2} c^{2} d x^{9} + \frac {1}{3} \, C b^{2} c d^{2} x^{9} + \frac {2}{9} \, D a b d^{3} x^{9} + \frac {1}{9} \, B b^{2} d^{3} x^{9} + \frac {1}{8} \, D b^{2} c^{3} x^{8} + \frac {3}{8} \, C b^{2} c^{2} d x^{8} + \frac {3}{4} \, D a b c d^{2} x^{8} + \frac {3}{8} \, B b^{2} c d^{2} x^{8} + \frac {1}{4} \, C a b d^{3} x^{8} + \frac {1}{8} \, A b^{2} d^{3} x^{8} + \frac {1}{7} \, C b^{2} c^{3} x^{7} + \frac {6}{7} \, D a b c^{2} d x^{7} + \frac {3}{7} \, B b^{2} c^{2} d x^{7} + \frac {6}{7} \, C a b c d^{2} x^{7} + \frac {3}{7} \, A b^{2} c d^{2} x^{7} + \frac {1}{7} \, D a^{2} d^{3} x^{7} + \frac {2}{7} \, B a b d^{3} x^{7} + \frac {1}{3} \, D a b c^{3} x^{6} + \frac {1}{6} \, B b^{2} c^{3} x^{6} + C a b c^{2} d x^{6} + \frac {1}{2} \, A b^{2} c^{2} d x^{6} + \frac {1}{2} \, D a^{2} c d^{2} x^{6} + B a b c d^{2} x^{6} + \frac {1}{6} \, C a^{2} d^{3} x^{6} + \frac {1}{3} \, A a b d^{3} x^{6} + \frac {2}{5} \, C a b c^{3} x^{5} + \frac {1}{5} \, A b^{2} c^{3} x^{5} + \frac {3}{5} \, D a^{2} c^{2} d x^{5} + \frac {6}{5} \, B a b c^{2} d x^{5} + \frac {3}{5} \, C a^{2} c d^{2} x^{5} + \frac {6}{5} \, A a b c d^{2} x^{5} + \frac {1}{5} \, B a^{2} d^{3} x^{5} + \frac {1}{4} \, D a^{2} c^{3} x^{4} + \frac {1}{2} \, B a b c^{3} x^{4} + \frac {3}{4} \, C a^{2} c^{2} d x^{4} + \frac {3}{2} \, A a b c^{2} d x^{4} + \frac {3}{4} \, B a^{2} c d^{2} x^{4} + \frac {1}{4} \, A a^{2} d^{3} x^{4} + \frac {1}{3} \, C a^{2} c^{3} x^{3} + \frac {2}{3} \, A a b c^{3} x^{3} + B a^{2} c^{2} d x^{3} + A a^{2} c d^{2} x^{3} + \frac {1}{2} \, B a^{2} c^{3} x^{2} + \frac {3}{2} \, A a^{2} c^{2} d x^{2} + A a^{2} c^{3} x \] Input:

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

Output:

1/11*D*b^2*d^3*x^11 + 3/10*D*b^2*c*d^2*x^10 + 1/10*C*b^2*d^3*x^10 + 1/3*D* 
b^2*c^2*d*x^9 + 1/3*C*b^2*c*d^2*x^9 + 2/9*D*a*b*d^3*x^9 + 1/9*B*b^2*d^3*x^ 
9 + 1/8*D*b^2*c^3*x^8 + 3/8*C*b^2*c^2*d*x^8 + 3/4*D*a*b*c*d^2*x^8 + 3/8*B* 
b^2*c*d^2*x^8 + 1/4*C*a*b*d^3*x^8 + 1/8*A*b^2*d^3*x^8 + 1/7*C*b^2*c^3*x^7 
+ 6/7*D*a*b*c^2*d*x^7 + 3/7*B*b^2*c^2*d*x^7 + 6/7*C*a*b*c*d^2*x^7 + 3/7*A* 
b^2*c*d^2*x^7 + 1/7*D*a^2*d^3*x^7 + 2/7*B*a*b*d^3*x^7 + 1/3*D*a*b*c^3*x^6 
+ 1/6*B*b^2*c^3*x^6 + C*a*b*c^2*d*x^6 + 1/2*A*b^2*c^2*d*x^6 + 1/2*D*a^2*c* 
d^2*x^6 + B*a*b*c*d^2*x^6 + 1/6*C*a^2*d^3*x^6 + 1/3*A*a*b*d^3*x^6 + 2/5*C* 
a*b*c^3*x^5 + 1/5*A*b^2*c^3*x^5 + 3/5*D*a^2*c^2*d*x^5 + 6/5*B*a*b*c^2*d*x^ 
5 + 3/5*C*a^2*c*d^2*x^5 + 6/5*A*a*b*c*d^2*x^5 + 1/5*B*a^2*d^3*x^5 + 1/4*D* 
a^2*c^3*x^4 + 1/2*B*a*b*c^3*x^4 + 3/4*C*a^2*c^2*d*x^4 + 3/2*A*a*b*c^2*d*x^ 
4 + 3/4*B*a^2*c*d^2*x^4 + 1/4*A*a^2*d^3*x^4 + 1/3*C*a^2*c^3*x^3 + 2/3*A*a* 
b*c^3*x^3 + B*a^2*c^2*d*x^3 + A*a^2*c*d^2*x^3 + 1/2*B*a^2*c^3*x^2 + 3/2*A* 
a^2*c^2*d*x^2 + A*a^2*c^3*x
 

Mupad [B] (verification not implemented)

Time = 22.12 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.27 \[ \int (c+d x)^3 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {a^2\,c^3\,x^4\,D}{4}+\frac {a^2\,d^3\,x^7\,D}{7}+\frac {b^2\,c^3\,x^8\,D}{8}+\frac {b^2\,d^3\,x^{11}\,D}{11}+A\,a^2\,c^3\,x+\frac {B\,a^2\,c^3\,x^2}{2}+\frac {A\,a^2\,d^3\,x^4}{4}+\frac {A\,b^2\,c^3\,x^5}{5}+\frac {C\,a^2\,c^3\,x^3}{3}+\frac {B\,a^2\,d^3\,x^5}{5}+\frac {B\,b^2\,c^3\,x^6}{6}+\frac {A\,b^2\,d^3\,x^8}{8}+\frac {C\,a^2\,d^3\,x^6}{6}+\frac {C\,b^2\,c^3\,x^7}{7}+\frac {B\,b^2\,d^3\,x^9}{9}+\frac {C\,b^2\,d^3\,x^{10}}{10}+\frac {C\,a\,b\,d^3\,x^8}{4}+\frac {a\,b\,c^3\,x^6\,D}{3}+\frac {2\,a\,b\,d^3\,x^9\,D}{9}+\frac {3\,A\,a^2\,c^2\,d\,x^2}{2}+A\,a^2\,c\,d^2\,x^3+B\,a^2\,c^2\,d\,x^3+\frac {3\,B\,a^2\,c\,d^2\,x^4}{4}+\frac {A\,b^2\,c^2\,d\,x^6}{2}+\frac {3\,A\,b^2\,c\,d^2\,x^7}{7}+\frac {3\,C\,a^2\,c^2\,d\,x^4}{4}+\frac {3\,C\,a^2\,c\,d^2\,x^5}{5}+\frac {3\,B\,b^2\,c^2\,d\,x^7}{7}+\frac {3\,B\,b^2\,c\,d^2\,x^8}{8}+\frac {3\,C\,b^2\,c^2\,d\,x^8}{8}+\frac {C\,b^2\,c\,d^2\,x^9}{3}+\frac {3\,a^2\,c^2\,d\,x^5\,D}{5}+\frac {a^2\,c\,d^2\,x^6\,D}{2}+\frac {b^2\,c^2\,d\,x^9\,D}{3}+\frac {3\,b^2\,c\,d^2\,x^{10}\,D}{10}+\frac {2\,A\,a\,b\,c^3\,x^3}{3}+\frac {B\,a\,b\,c^3\,x^4}{2}+\frac {A\,a\,b\,d^3\,x^6}{3}+\frac {2\,C\,a\,b\,c^3\,x^5}{5}+\frac {2\,B\,a\,b\,d^3\,x^7}{7}+\frac {6\,a\,b\,c^2\,d\,x^7\,D}{7}+\frac {3\,a\,b\,c\,d^2\,x^8\,D}{4}+\frac {3\,A\,a\,b\,c^2\,d\,x^4}{2}+\frac {6\,A\,a\,b\,c\,d^2\,x^5}{5}+\frac {6\,B\,a\,b\,c^2\,d\,x^5}{5}+B\,a\,b\,c\,d^2\,x^6+C\,a\,b\,c^2\,d\,x^6+\frac {6\,C\,a\,b\,c\,d^2\,x^7}{7} \] Input:

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

Output:

(a^2*c^3*x^4*D)/4 + (a^2*d^3*x^7*D)/7 + (b^2*c^3*x^8*D)/8 + (b^2*d^3*x^11* 
D)/11 + A*a^2*c^3*x + (B*a^2*c^3*x^2)/2 + (A*a^2*d^3*x^4)/4 + (A*b^2*c^3*x 
^5)/5 + (C*a^2*c^3*x^3)/3 + (B*a^2*d^3*x^5)/5 + (B*b^2*c^3*x^6)/6 + (A*b^2 
*d^3*x^8)/8 + (C*a^2*d^3*x^6)/6 + (C*b^2*c^3*x^7)/7 + (B*b^2*d^3*x^9)/9 + 
(C*b^2*d^3*x^10)/10 + (C*a*b*d^3*x^8)/4 + (a*b*c^3*x^6*D)/3 + (2*a*b*d^3*x 
^9*D)/9 + (3*A*a^2*c^2*d*x^2)/2 + A*a^2*c*d^2*x^3 + B*a^2*c^2*d*x^3 + (3*B 
*a^2*c*d^2*x^4)/4 + (A*b^2*c^2*d*x^6)/2 + (3*A*b^2*c*d^2*x^7)/7 + (3*C*a^2 
*c^2*d*x^4)/4 + (3*C*a^2*c*d^2*x^5)/5 + (3*B*b^2*c^2*d*x^7)/7 + (3*B*b^2*c 
*d^2*x^8)/8 + (3*C*b^2*c^2*d*x^8)/8 + (C*b^2*c*d^2*x^9)/3 + (3*a^2*c^2*d*x 
^5*D)/5 + (a^2*c*d^2*x^6*D)/2 + (b^2*c^2*d*x^9*D)/3 + (3*b^2*c*d^2*x^10*D) 
/10 + (2*A*a*b*c^3*x^3)/3 + (B*a*b*c^3*x^4)/2 + (A*a*b*d^3*x^6)/3 + (2*C*a 
*b*c^3*x^5)/5 + (2*B*a*b*d^3*x^7)/7 + (6*a*b*c^2*d*x^7*D)/7 + (3*a*b*c*d^2 
*x^8*D)/4 + (3*A*a*b*c^2*d*x^4)/2 + (6*A*a*b*c*d^2*x^5)/5 + (6*B*a*b*c^2*d 
*x^5)/5 + B*a*b*c*d^2*x^6 + C*a*b*c^2*d*x^6 + (6*C*a*b*c*d^2*x^7)/7
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.02 \[ \int (c+d x)^3 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {x \left (2520 b^{2} d^{4} x^{10}+11088 b^{2} c \,d^{3} x^{9}+6160 a b \,d^{4} x^{8}+3080 b^{3} d^{3} x^{8}+18480 b^{2} c^{2} d^{2} x^{8}+3465 a \,b^{2} d^{3} x^{7}+27720 a b c \,d^{3} x^{7}+10395 b^{3} c \,d^{2} x^{7}+13860 b^{2} c^{3} d \,x^{7}+3960 a^{2} d^{4} x^{6}+11880 a \,b^{2} c \,d^{2} x^{6}+7920 a \,b^{2} d^{3} x^{6}+47520 a b \,c^{2} d^{2} x^{6}+11880 b^{3} c^{2} d \,x^{6}+3960 b^{2} c^{4} x^{6}+9240 a^{2} b \,d^{3} x^{5}+18480 a^{2} c \,d^{3} x^{5}+13860 a \,b^{2} c^{2} d \,x^{5}+27720 a \,b^{2} c \,d^{2} x^{5}+36960 a b \,c^{3} d \,x^{5}+4620 b^{3} c^{3} x^{5}+33264 a^{2} b c \,d^{2} x^{4}+5544 a^{2} b \,d^{3} x^{4}+33264 a^{2} c^{2} d^{2} x^{4}+5544 a \,b^{2} c^{3} x^{4}+33264 a \,b^{2} c^{2} d \,x^{4}+11088 a b \,c^{4} x^{4}+6930 a^{3} d^{3} x^{3}+41580 a^{2} b \,c^{2} d \,x^{3}+20790 a^{2} b c \,d^{2} x^{3}+27720 a^{2} c^{3} d \,x^{3}+13860 a \,b^{2} c^{3} x^{3}+27720 a^{3} c \,d^{2} x^{2}+18480 a^{2} b \,c^{3} x^{2}+27720 a^{2} b \,c^{2} d \,x^{2}+9240 a^{2} c^{4} x^{2}+41580 a^{3} c^{2} d x +13860 a^{2} b \,c^{3} x +27720 a^{3} c^{3}\right )}{27720} \] Input:

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

Output:

(x*(27720*a**3*c**3 + 41580*a**3*c**2*d*x + 27720*a**3*c*d**2*x**2 + 6930* 
a**3*d**3*x**3 + 18480*a**2*b*c**3*x**2 + 13860*a**2*b*c**3*x + 41580*a**2 
*b*c**2*d*x**3 + 27720*a**2*b*c**2*d*x**2 + 33264*a**2*b*c*d**2*x**4 + 207 
90*a**2*b*c*d**2*x**3 + 9240*a**2*b*d**3*x**5 + 5544*a**2*b*d**3*x**4 + 92 
40*a**2*c**4*x**2 + 27720*a**2*c**3*d*x**3 + 33264*a**2*c**2*d**2*x**4 + 1 
8480*a**2*c*d**3*x**5 + 3960*a**2*d**4*x**6 + 5544*a*b**2*c**3*x**4 + 1386 
0*a*b**2*c**3*x**3 + 13860*a*b**2*c**2*d*x**5 + 33264*a*b**2*c**2*d*x**4 + 
 11880*a*b**2*c*d**2*x**6 + 27720*a*b**2*c*d**2*x**5 + 3465*a*b**2*d**3*x* 
*7 + 7920*a*b**2*d**3*x**6 + 11088*a*b*c**4*x**4 + 36960*a*b*c**3*d*x**5 + 
 47520*a*b*c**2*d**2*x**6 + 27720*a*b*c*d**3*x**7 + 6160*a*b*d**4*x**8 + 4 
620*b**3*c**3*x**5 + 11880*b**3*c**2*d*x**6 + 10395*b**3*c*d**2*x**7 + 308 
0*b**3*d**3*x**8 + 3960*b**2*c**4*x**6 + 13860*b**2*c**3*d*x**7 + 18480*b* 
*2*c**2*d**2*x**8 + 11088*b**2*c*d**3*x**9 + 2520*b**2*d**4*x**10))/27720