\(\int (c+d x)^2 (a+b x^2)^3 (A+B x+C x^2+D x^3) \, dx\) [16]

Optimal result

Integrand size = 32, antiderivative size = 418 \[ \int (c+d x)^2 \left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right ) \, dx=a^3 A c^2 x+\frac {1}{2} a^3 c (B c+2 A d) x^2+\frac {1}{3} a^2 \left (a c (c C+2 B d)+A \left (3 b c^2+a d^2\right )\right ) x^3+\frac {1}{4} a^2 \left (3 b c (B c+2 A d)+a \left (2 c C d+B d^2+c^2 D\right )\right ) x^4+\frac {1}{5} a \left (3 A b \left (b c^2+a d^2\right )+a (3 b c (c C+2 B d)+a d (C d+2 c D))\right ) x^5+\frac {1}{6} a \left (3 b^2 c (B c+2 A d)+a^2 d^2 D+3 a b \left (2 c C d+B d^2+c^2 D\right )\right ) x^6+\frac {1}{7} b \left (A b \left (b c^2+3 a d^2\right )+3 a (b c (c C+2 B d)+a d (C d+2 c D))\right ) x^7+\frac {1}{8} b \left (b^2 c (B c+2 A d)+3 a^2 d^2 D+3 a b \left (2 c C d+B d^2+c^2 D\right )\right ) x^8+\frac {1}{9} b^2 \left (b \left (c^2 C+2 B c d+A d^2\right )+3 a d (C d+2 c D)\right ) x^9+\frac {1}{10} b^2 \left (3 a d^2 D+b \left (2 c C d+B d^2+c^2 D\right )\right ) x^{10}+\frac {1}{11} b^3 d (C d+2 c D) x^{11}+\frac {1}{12} b^3 d^2 D x^{12} \] Output:

a^3*A*c^2*x+1/2*a^3*c*(2*A*d+B*c)*x^2+1/3*a^2*(a*c*(2*B*d+C*c)+A*(a*d^2+3* 
b*c^2))*x^3+1/4*a^2*(3*b*c*(2*A*d+B*c)+a*(B*d^2+2*C*c*d+D*c^2))*x^4+1/5*a* 
(3*A*b*(a*d^2+b*c^2)+a*(3*b*c*(2*B*d+C*c)+a*d*(C*d+2*D*c)))*x^5+1/6*a*(3*b 
^2*c*(2*A*d+B*c)+a^2*d^2*D+3*a*b*(B*d^2+2*C*c*d+D*c^2))*x^6+1/7*b*(A*b*(3* 
a*d^2+b*c^2)+3*a*(b*c*(2*B*d+C*c)+a*d*(C*d+2*D*c)))*x^7+1/8*b*(b^2*c*(2*A* 
d+B*c)+3*a^2*d^2*D+3*a*b*(B*d^2+2*C*c*d+D*c^2))*x^8+1/9*b^2*(b*(A*d^2+2*B* 
c*d+C*c^2)+3*a*d*(C*d+2*D*c))*x^9+1/10*b^2*(3*a*d^2*D+b*(B*d^2+2*C*c*d+D*c 
^2))*x^10+1/11*b^3*d*(C*d+2*D*c)*x^11+1/12*b^3*d^2*D*x^12
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.00 \[ \int (c+d x)^2 \left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right ) \, dx=a^3 A c^2 x+\frac {1}{2} a^3 c (B c+2 A d) x^2+\frac {1}{3} a^2 \left (a c (c C+2 B d)+A \left (3 b c^2+a d^2\right )\right ) x^3+\frac {1}{4} a^2 \left (3 b c (B c+2 A d)+a \left (2 c C d+B d^2+c^2 D\right )\right ) x^4+\frac {1}{5} a \left (3 A b \left (b c^2+a d^2\right )+a (3 b c (c C+2 B d)+a d (C d+2 c D))\right ) x^5+\frac {1}{6} a \left (3 b^2 c (B c+2 A d)+a^2 d^2 D+3 a b \left (2 c C d+B d^2+c^2 D\right )\right ) x^6+\frac {1}{7} b \left (A b \left (b c^2+3 a d^2\right )+3 a (b c (c C+2 B d)+a d (C d+2 c D))\right ) x^7+\frac {1}{8} b \left (b^2 c (B c+2 A d)+3 a^2 d^2 D+3 a b \left (2 c C d+B d^2+c^2 D\right )\right ) x^8+\frac {1}{9} b^2 \left (b \left (c^2 C+2 B c d+A d^2\right )+3 a d (C d+2 c D)\right ) x^9+\frac {1}{10} b^2 \left (3 a d^2 D+b \left (2 c C d+B d^2+c^2 D\right )\right ) x^{10}+\frac {1}{11} b^3 d (C d+2 c D) x^{11}+\frac {1}{12} b^3 d^2 D x^{12} \] Input:

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

Output:

a^3*A*c^2*x + (a^3*c*(B*c + 2*A*d)*x^2)/2 + (a^2*(a*c*(c*C + 2*B*d) + A*(3 
*b*c^2 + a*d^2))*x^3)/3 + (a^2*(3*b*c*(B*c + 2*A*d) + a*(2*c*C*d + B*d^2 + 
 c^2*D))*x^4)/4 + (a*(3*A*b*(b*c^2 + a*d^2) + a*(3*b*c*(c*C + 2*B*d) + a*d 
*(C*d + 2*c*D)))*x^5)/5 + (a*(3*b^2*c*(B*c + 2*A*d) + a^2*d^2*D + 3*a*b*(2 
*c*C*d + B*d^2 + c^2*D))*x^6)/6 + (b*(A*b*(b*c^2 + 3*a*d^2) + 3*a*(b*c*(c* 
C + 2*B*d) + a*d*(C*d + 2*c*D)))*x^7)/7 + (b*(b^2*c*(B*c + 2*A*d) + 3*a^2* 
d^2*D + 3*a*b*(2*c*C*d + B*d^2 + c^2*D))*x^8)/8 + (b^2*(b*(c^2*C + 2*B*c*d 
 + A*d^2) + 3*a*d*(C*d + 2*c*D))*x^9)/9 + (b^2*(3*a*d^2*D + b*(2*c*C*d + B 
*d^2 + c^2*D))*x^10)/10 + (b^3*d*(C*d + 2*c*D)*x^11)/11 + (b^3*d^2*D*x^12) 
/12
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.90, 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.

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

\(\Big \downarrow \) 2017

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

\(\Big \downarrow \) 2341

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

\(\Big \downarrow \) 2009

\(\displaystyle a^3 A c^2 x+\frac {1}{4} a^3 x^4 \left (B d^2+c^2 D+2 c C d\right )+\frac {1}{3} a^2 x^3 \left (A \left (a d^2+3 b c^2\right )+a c (2 B d+c C)\right )+\frac {1}{6} a^2 x^6 \left (a d^2 D+3 b \left (B d^2+c^2 D+2 c C d\right )\right )+\frac {1}{9} b^2 x^9 \left (3 a d (2 c D+C d)+b \left (A d^2+2 B c d+c^2 C\right )\right )+\frac {1}{7} b x^7 \left (A b \left (3 a d^2+b c^2\right )+3 a (a d (2 c D+C d)+b c (2 B d+c C))\right )+\frac {1}{5} a x^5 \left (3 A b \left (a d^2+b c^2\right )+a (a d (2 c D+C d)+3 b c (2 B d+c C))\right )+\frac {c \left (a+b x^2\right )^4 (2 A d+B c)}{8 b}+\frac {1}{10} b^2 x^{10} \left (3 a d^2 D+b \left (B d^2+c^2 D+2 c C d\right )\right )+\frac {3}{8} a b x^8 \left (a d^2 D+b \left (B d^2+c^2 D+2 c C d\right )\right )+\frac {1}{11} b^3 d x^{11} (2 c D+C d)+\frac {1}{12} b^3 d^2 D x^{12}\)

Input:

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

Output:

a^3*A*c^2*x + (a^2*(a*c*(c*C + 2*B*d) + A*(3*b*c^2 + a*d^2))*x^3)/3 + (a^3 
*(2*c*C*d + B*d^2 + c^2*D)*x^4)/4 + (a*(3*A*b*(b*c^2 + a*d^2) + a*(3*b*c*( 
c*C + 2*B*d) + a*d*(C*d + 2*c*D)))*x^5)/5 + (a^2*(a*d^2*D + 3*b*(2*c*C*d + 
 B*d^2 + c^2*D))*x^6)/6 + (b*(A*b*(b*c^2 + 3*a*d^2) + 3*a*(b*c*(c*C + 2*B* 
d) + a*d*(C*d + 2*c*D)))*x^7)/7 + (3*a*b*(a*d^2*D + b*(2*c*C*d + B*d^2 + c 
^2*D))*x^8)/8 + (b^2*(b*(c^2*C + 2*B*c*d + A*d^2) + 3*a*d*(C*d + 2*c*D))*x 
^9)/9 + (b^2*(3*a*d^2*D + b*(2*c*C*d + B*d^2 + c^2*D))*x^10)/10 + (b^3*d*( 
C*d + 2*c*D)*x^11)/11 + (b^3*d^2*D*x^12)/12 + (c*(B*c + 2*A*d)*(a + b*x^2) 
^4)/(8*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 = 0.75 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.17

method	result	size
norman	\(\frac {b^{3} d^{2} D x^{12}}{12}+\left (\frac {1}{11} b^{3} d^{2} C +\frac {2}{11} b^{3} c d D\right ) x^{11}+\left (\frac {1}{10} B \,b^{3} d^{2}+\frac {1}{5} b^{3} c d C +\frac {3}{10} D a \,b^{2} d^{2}+\frac {1}{10} D b^{3} c^{2}\right ) x^{10}+\left (\frac {1}{9} A \,b^{3} d^{2}+\frac {2}{9} b^{3} c d B +\frac {1}{3} C a \,b^{2} d^{2}+\frac {1}{9} C \,b^{3} c^{2}+\frac {2}{3} c d a \,b^{2} D\right ) x^{9}+\left (\frac {1}{4} b^{3} c d A +\frac {3}{8} B a \,b^{2} d^{2}+\frac {1}{8} B \,b^{3} c^{2}+\frac {3}{4} c d a \,b^{2} C +\frac {3}{8} D a^{2} b \,d^{2}+\frac {3}{8} D a \,b^{2} c^{2}\right ) x^{8}+\left (\frac {3}{7} A \,d^{2} a \,b^{2}+\frac {1}{7} A \,b^{3} c^{2}+\frac {6}{7} c d a \,b^{2} B +\frac {3}{7} C \,a^{2} b \,d^{2}+\frac {3}{7} C a \,b^{2} c^{2}+\frac {6}{7} a^{2} b c d D\right ) x^{7}+\left (c d a \,b^{2} A +\frac {1}{2} B \,a^{2} b \,d^{2}+\frac {1}{2} B a \,b^{2} c^{2}+a^{2} b c d C +\frac {1}{6} D a^{3} d^{2}+\frac {1}{2} D a^{2} b \,c^{2}\right ) x^{6}+\left (\frac {3}{5} A \,d^{2} a^{2} b +\frac {3}{5} A a \,b^{2} c^{2}+\frac {6}{5} a^{2} b c d B +\frac {1}{5} C \,a^{3} d^{2}+\frac {3}{5} C \,a^{2} b \,c^{2}+\frac {2}{5} a^{3} d c D\right ) x^{5}+\left (\frac {3}{2} a^{2} b c d A +\frac {1}{4} B \,a^{3} d^{2}+\frac {3}{4} B \,a^{2} b \,c^{2}+\frac {1}{2} a^{3} d c C +\frac {1}{4} c^{2} a^{3} D\right ) x^{4}+\left (\frac {1}{3} A \,d^{2} a^{3}+A \,a^{2} b \,c^{2}+\frac {2}{3} a^{3} d c B +\frac {1}{3} c^{2} a^{3} C \right ) x^{3}+\left (a^{3} d c A +\frac {1}{2} B \,a^{3} c^{2}\right ) x^{2}+a^{3} A \,c^{2} x\)	\(491\)
default	\(\frac {b^{3} d^{2} D x^{12}}{12}+\frac {\left (b^{3} d^{2} C +2 b^{3} c d D\right ) x^{11}}{11}+\frac {\left (\left (3 a \,b^{2} d^{2}+b^{3} c^{2}\right ) D+2 b^{3} c d C +B \,b^{3} d^{2}\right ) x^{10}}{10}+\frac {\left (6 c d a \,b^{2} D+\left (3 a \,b^{2} d^{2}+b^{3} c^{2}\right ) C +2 b^{3} c d B +A \,b^{3} d^{2}\right ) x^{9}}{9}+\frac {\left (\left (3 a^{2} b \,d^{2}+3 a \,c^{2} b^{2}\right ) D+6 c d a \,b^{2} C +\left (3 a \,b^{2} d^{2}+b^{3} c^{2}\right ) B +2 b^{3} c d A \right ) x^{8}}{8}+\frac {\left (6 a^{2} b c d D+\left (3 a^{2} b \,d^{2}+3 a \,c^{2} b^{2}\right ) C +6 c d a \,b^{2} B +\left (3 a \,b^{2} d^{2}+b^{3} c^{2}\right ) A \right ) x^{7}}{7}+\frac {\left (\left (a^{3} d^{2}+3 a^{2} b \,c^{2}\right ) D+6 a^{2} b c d C +\left (3 a^{2} b \,d^{2}+3 a \,c^{2} b^{2}\right ) B +6 c d a \,b^{2} A \right ) x^{6}}{6}+\frac {\left (2 a^{3} d c D+\left (a^{3} d^{2}+3 a^{2} b \,c^{2}\right ) C +6 a^{2} b c d B +\left (3 a^{2} b \,d^{2}+3 a \,c^{2} b^{2}\right ) A \right ) x^{5}}{5}+\frac {\left (c^{2} a^{3} D+2 a^{3} d c C +\left (a^{3} d^{2}+3 a^{2} b \,c^{2}\right ) B +6 a^{2} b c d A \right ) x^{4}}{4}+\frac {\left (c^{2} a^{3} C +2 a^{3} d c B +\left (a^{3} d^{2}+3 a^{2} b \,c^{2}\right ) A \right ) x^{3}}{3}+\frac {\left (2 a^{3} d c A +B \,a^{3} c^{2}\right ) x^{2}}{2}+a^{3} A \,c^{2} x\)	\(503\)
gosper	\(\frac {1}{2} x^{2} B \,a^{3} c^{2}+\frac {1}{5} x^{10} b^{3} c d C +\frac {3}{5} x^{5} A \,d^{2} a^{2} b +\frac {1}{3} x^{9} C a \,b^{2} d^{2}+\frac {1}{9} x^{9} C \,b^{3} c^{2}+\frac {1}{2} x^{4} a^{3} d c C +x^{2} a^{3} d c A +\frac {1}{5} x^{5} C \,a^{3} d^{2}+\frac {1}{2} x^{6} B \,a^{2} b \,d^{2}+\frac {3}{10} x^{10} D a \,b^{2} d^{2}+\frac {3}{4} x^{8} c d a \,b^{2} C +\frac {6}{7} x^{7} c d a \,b^{2} B +\frac {6}{7} x^{7} a^{2} b c d D+x^{6} c d a \,b^{2} A +\frac {1}{8} x^{8} B \,b^{3} c^{2}+\frac {1}{7} x^{7} A \,b^{3} c^{2}+\frac {1}{6} x^{6} D a^{3} d^{2}+\frac {3}{8} x^{8} D a^{2} b \,d^{2}+\frac {3}{7} x^{7} C a \,b^{2} c^{2}+\frac {2}{9} x^{9} b^{3} c d B +\frac {1}{4} x^{8} b^{3} c d A +\frac {3}{8} x^{8} D a \,b^{2} c^{2}+\frac {1}{2} x^{6} D a^{2} b \,c^{2}+x^{3} A \,a^{2} b \,c^{2}+\frac {2}{3} x^{3} a^{3} d c B +\frac {2}{11} x^{11} b^{3} c d D+\frac {3}{4} x^{4} B \,a^{2} b \,c^{2}+\frac {3}{5} x^{5} A a \,b^{2} c^{2}+\frac {3}{5} x^{5} C \,a^{2} b \,c^{2}+\frac {2}{5} x^{5} a^{3} d c D+\frac {3}{8} x^{8} B a \,b^{2} d^{2}+\frac {3}{7} x^{7} A \,d^{2} a \,b^{2}+\frac {3}{7} x^{7} C \,a^{2} b \,d^{2}+x^{6} a^{2} b c d C +\frac {6}{5} x^{5} a^{2} b c d B +\frac {3}{2} x^{4} a^{2} b c d A +\frac {1}{2} x^{6} B a \,b^{2} c^{2}+\frac {1}{11} x^{11} b^{3} d^{2} C +\frac {1}{4} x^{4} B \,a^{3} d^{2}+\frac {1}{4} x^{4} c^{2} a^{3} D+\frac {1}{3} x^{3} A \,d^{2} a^{3}+\frac {1}{10} x^{10} B \,b^{3} d^{2}+\frac {1}{10} x^{10} D b^{3} c^{2}+\frac {1}{9} x^{9} A \,b^{3} d^{2}+\frac {1}{3} x^{3} c^{2} a^{3} C +\frac {2}{3} x^{9} c d a \,b^{2} D+a^{3} A \,c^{2} x +\frac {1}{12} b^{3} d^{2} D x^{12}\)	\(579\)
parallelrisch	\(\frac {1}{2} x^{2} B \,a^{3} c^{2}+\frac {1}{5} x^{10} b^{3} c d C +\frac {3}{5} x^{5} A \,d^{2} a^{2} b +\frac {1}{3} x^{9} C a \,b^{2} d^{2}+\frac {1}{9} x^{9} C \,b^{3} c^{2}+\frac {1}{2} x^{4} a^{3} d c C +x^{2} a^{3} d c A +\frac {1}{5} x^{5} C \,a^{3} d^{2}+\frac {1}{2} x^{6} B \,a^{2} b \,d^{2}+\frac {3}{10} x^{10} D a \,b^{2} d^{2}+\frac {3}{4} x^{8} c d a \,b^{2} C +\frac {6}{7} x^{7} c d a \,b^{2} B +\frac {6}{7} x^{7} a^{2} b c d D+x^{6} c d a \,b^{2} A +\frac {1}{8} x^{8} B \,b^{3} c^{2}+\frac {1}{7} x^{7} A \,b^{3} c^{2}+\frac {1}{6} x^{6} D a^{3} d^{2}+\frac {3}{8} x^{8} D a^{2} b \,d^{2}+\frac {3}{7} x^{7} C a \,b^{2} c^{2}+\frac {2}{9} x^{9} b^{3} c d B +\frac {1}{4} x^{8} b^{3} c d A +\frac {3}{8} x^{8} D a \,b^{2} c^{2}+\frac {1}{2} x^{6} D a^{2} b \,c^{2}+x^{3} A \,a^{2} b \,c^{2}+\frac {2}{3} x^{3} a^{3} d c B +\frac {2}{11} x^{11} b^{3} c d D+\frac {3}{4} x^{4} B \,a^{2} b \,c^{2}+\frac {3}{5} x^{5} A a \,b^{2} c^{2}+\frac {3}{5} x^{5} C \,a^{2} b \,c^{2}+\frac {2}{5} x^{5} a^{3} d c D+\frac {3}{8} x^{8} B a \,b^{2} d^{2}+\frac {3}{7} x^{7} A \,d^{2} a \,b^{2}+\frac {3}{7} x^{7} C \,a^{2} b \,d^{2}+x^{6} a^{2} b c d C +\frac {6}{5} x^{5} a^{2} b c d B +\frac {3}{2} x^{4} a^{2} b c d A +\frac {1}{2} x^{6} B a \,b^{2} c^{2}+\frac {1}{11} x^{11} b^{3} d^{2} C +\frac {1}{4} x^{4} B \,a^{3} d^{2}+\frac {1}{4} x^{4} c^{2} a^{3} D+\frac {1}{3} x^{3} A \,d^{2} a^{3}+\frac {1}{10} x^{10} B \,b^{3} d^{2}+\frac {1}{10} x^{10} D b^{3} c^{2}+\frac {1}{9} x^{9} A \,b^{3} d^{2}+\frac {1}{3} x^{3} c^{2} a^{3} C +\frac {2}{3} x^{9} c d a \,b^{2} D+a^{3} A \,c^{2} x +\frac {1}{12} b^{3} d^{2} D x^{12}\)	\(579\)
orering	\(\frac {x \left (2310 b^{3} d^{2} D x^{11}+2520 C \,b^{3} d^{2} x^{10}+5040 D b^{3} c d \,x^{10}+2772 B \,b^{3} d^{2} x^{9}+5544 C \,b^{3} c d \,x^{9}+8316 D a \,b^{2} d^{2} x^{9}+2772 D b^{3} c^{2} x^{9}+3080 A \,b^{3} d^{2} x^{8}+6160 B \,b^{3} c d \,x^{8}+9240 C a \,b^{2} d^{2} x^{8}+3080 C \,b^{3} c^{2} x^{8}+18480 D a \,b^{2} c d \,x^{8}+6930 A \,b^{3} c d \,x^{7}+10395 B a \,b^{2} d^{2} x^{7}+3465 B \,b^{3} c^{2} x^{7}+20790 C a \,b^{2} c d \,x^{7}+10395 D a^{2} b \,d^{2} x^{7}+10395 D a \,b^{2} c^{2} x^{7}+11880 A a \,b^{2} d^{2} x^{6}+3960 A \,b^{3} c^{2} x^{6}+23760 B a \,b^{2} c d \,x^{6}+11880 C \,a^{2} b \,d^{2} x^{6}+11880 C a \,b^{2} c^{2} x^{6}+23760 D a^{2} b c d \,x^{6}+27720 A a \,b^{2} c d \,x^{5}+13860 B \,a^{2} b \,d^{2} x^{5}+13860 B a \,b^{2} c^{2} x^{5}+27720 C \,a^{2} b c d \,x^{5}+4620 D a^{3} d^{2} x^{5}+13860 D a^{2} b \,c^{2} x^{5}+16632 A \,a^{2} b \,d^{2} x^{4}+16632 A a \,b^{2} c^{2} x^{4}+33264 B \,a^{2} b c d \,x^{4}+5544 C \,a^{3} d^{2} x^{4}+16632 C \,a^{2} b \,c^{2} x^{4}+11088 D a^{3} c d \,x^{4}+41580 A \,a^{2} b c d \,x^{3}+6930 B \,a^{3} d^{2} x^{3}+20790 B \,a^{2} b \,c^{2} x^{3}+13860 C \,a^{3} c d \,x^{3}+6930 D a^{3} c^{2} x^{3}+9240 A \,a^{3} d^{2} x^{2}+27720 A \,a^{2} b \,c^{2} x^{2}+18480 B \,a^{3} c d \,x^{2}+9240 C \,a^{3} c^{2} x^{2}+27720 A \,a^{3} c d x +13860 B \,a^{3} c^{2} x +27720 a^{3} A \,c^{2}\right )}{27720}\)	\(582\)

Input:

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

Output:

1/12*b^3*d^2*D*x^12+(1/11*b^3*d^2*C+2/11*b^3*c*d*D)*x^11+(1/10*B*b^3*d^2+1 
/5*b^3*c*d*C+3/10*D*a*b^2*d^2+1/10*D*b^3*c^2)*x^10+(1/9*A*b^3*d^2+2/9*b^3* 
c*d*B+1/3*C*a*b^2*d^2+1/9*C*b^3*c^2+2/3*c*d*a*b^2*D)*x^9+(1/4*b^3*c*d*A+3/ 
8*B*a*b^2*d^2+1/8*B*b^3*c^2+3/4*c*d*a*b^2*C+3/8*D*a^2*b*d^2+3/8*D*a*b^2*c^ 
2)*x^8+(3/7*A*d^2*a*b^2+1/7*A*b^3*c^2+6/7*c*d*a*b^2*B+3/7*C*a^2*b*d^2+3/7* 
C*a*b^2*c^2+6/7*a^2*b*c*d*D)*x^7+(c*d*a*b^2*A+1/2*B*a^2*b*d^2+1/2*B*a*b^2* 
c^2+a^2*b*c*d*C+1/6*D*a^3*d^2+1/2*D*a^2*b*c^2)*x^6+(3/5*A*d^2*a^2*b+3/5*A* 
a*b^2*c^2+6/5*a^2*b*c*d*B+1/5*C*a^3*d^2+3/5*C*a^2*b*c^2+2/5*a^3*d*c*D)*x^5 
+(3/2*a^2*b*c*d*A+1/4*B*a^3*d^2+3/4*B*a^2*b*c^2+1/2*a^3*d*c*C+1/4*c^2*a^3* 
D)*x^4+(1/3*A*d^2*a^3+A*a^2*b*c^2+2/3*a^3*d*c*B+1/3*c^2*a^3*C)*x^3+(a^3*d* 
c*A+1/2*B*a^3*c^2)*x^2+a^3*A*c^2*x
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.13 \[ \int (c+d x)^2 \left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{12} \, D b^{3} d^{2} x^{12} + \frac {1}{11} \, {\left (2 \, D b^{3} c d + C b^{3} d^{2}\right )} x^{11} + \frac {1}{10} \, {\left (D b^{3} c^{2} + 2 \, C b^{3} c d + {\left (3 \, D a b^{2} + B b^{3}\right )} d^{2}\right )} x^{10} + \frac {1}{9} \, {\left (C b^{3} c^{2} + 2 \, {\left (3 \, D a b^{2} + B b^{3}\right )} c d + {\left (3 \, C a b^{2} + A b^{3}\right )} d^{2}\right )} x^{9} + \frac {1}{8} \, {\left ({\left (3 \, D a b^{2} + B b^{3}\right )} c^{2} + 2 \, {\left (3 \, C a b^{2} + A b^{3}\right )} c d + 3 \, {\left (D a^{2} b + B a b^{2}\right )} d^{2}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, C a b^{2} + A b^{3}\right )} c^{2} + 6 \, {\left (D a^{2} b + B a b^{2}\right )} c d + 3 \, {\left (C a^{2} b + A a b^{2}\right )} d^{2}\right )} x^{7} + A a^{3} c^{2} x + \frac {1}{6} \, {\left (3 \, {\left (D a^{2} b + B a b^{2}\right )} c^{2} + 6 \, {\left (C a^{2} b + A a b^{2}\right )} c d + {\left (D a^{3} + 3 \, B a^{2} b\right )} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, {\left (C a^{2} b + A a b^{2}\right )} c^{2} + 2 \, {\left (D a^{3} + 3 \, B a^{2} b\right )} c d + {\left (C a^{3} + 3 \, A a^{2} b\right )} d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{3} d^{2} + {\left (D a^{3} + 3 \, B a^{2} b\right )} c^{2} + 2 \, {\left (C a^{3} + 3 \, A a^{2} b\right )} c d\right )} x^{4} + \frac {1}{3} \, {\left (2 \, B a^{3} c d + A a^{3} d^{2} + {\left (C a^{3} + 3 \, A a^{2} b\right )} c^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} c^{2} + 2 \, A a^{3} c d\right )} x^{2} \] Input:

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

Output:

1/12*D*b^3*d^2*x^12 + 1/11*(2*D*b^3*c*d + C*b^3*d^2)*x^11 + 1/10*(D*b^3*c^ 
2 + 2*C*b^3*c*d + (3*D*a*b^2 + B*b^3)*d^2)*x^10 + 1/9*(C*b^3*c^2 + 2*(3*D* 
a*b^2 + B*b^3)*c*d + (3*C*a*b^2 + A*b^3)*d^2)*x^9 + 1/8*((3*D*a*b^2 + B*b^ 
3)*c^2 + 2*(3*C*a*b^2 + A*b^3)*c*d + 3*(D*a^2*b + B*a*b^2)*d^2)*x^8 + 1/7* 
((3*C*a*b^2 + A*b^3)*c^2 + 6*(D*a^2*b + B*a*b^2)*c*d + 3*(C*a^2*b + A*a*b^ 
2)*d^2)*x^7 + A*a^3*c^2*x + 1/6*(3*(D*a^2*b + B*a*b^2)*c^2 + 6*(C*a^2*b + 
A*a*b^2)*c*d + (D*a^3 + 3*B*a^2*b)*d^2)*x^6 + 1/5*(3*(C*a^2*b + A*a*b^2)*c 
^2 + 2*(D*a^3 + 3*B*a^2*b)*c*d + (C*a^3 + 3*A*a^2*b)*d^2)*x^5 + 1/4*(B*a^3 
*d^2 + (D*a^3 + 3*B*a^2*b)*c^2 + 2*(C*a^3 + 3*A*a^2*b)*c*d)*x^4 + 1/3*(2*B 
*a^3*c*d + A*a^3*d^2 + (C*a^3 + 3*A*a^2*b)*c^2)*x^3 + 1/2*(B*a^3*c^2 + 2*A 
*a^3*c*d)*x^2
 

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.42 \[ \int (c+d x)^2 \left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right ) \, dx=A a^{3} c^{2} x + \frac {D b^{3} d^{2} x^{12}}{12} + x^{11} \left (\frac {C b^{3} d^{2}}{11} + \frac {2 D b^{3} c d}{11}\right ) + x^{10} \left (\frac {B b^{3} d^{2}}{10} + \frac {C b^{3} c d}{5} + \frac {3 D a b^{2} d^{2}}{10} + \frac {D b^{3} c^{2}}{10}\right ) + x^{9} \left (\frac {A b^{3} d^{2}}{9} + \frac {2 B b^{3} c d}{9} + \frac {C a b^{2} d^{2}}{3} + \frac {C b^{3} c^{2}}{9} + \frac {2 D a b^{2} c d}{3}\right ) + x^{8} \left (\frac {A b^{3} c d}{4} + \frac {3 B a b^{2} d^{2}}{8} + \frac {B b^{3} c^{2}}{8} + \frac {3 C a b^{2} c d}{4} + \frac {3 D a^{2} b d^{2}}{8} + \frac {3 D a b^{2} c^{2}}{8}\right ) + x^{7} \cdot \left (\frac {3 A a b^{2} d^{2}}{7} + \frac {A b^{3} c^{2}}{7} + \frac {6 B a b^{2} c d}{7} + \frac {3 C a^{2} b d^{2}}{7} + \frac {3 C a b^{2} c^{2}}{7} + \frac {6 D a^{2} b c d}{7}\right ) + x^{6} \left (A a b^{2} c d + \frac {B a^{2} b d^{2}}{2} + \frac {B a b^{2} c^{2}}{2} + C a^{2} b c d + \frac {D a^{3} d^{2}}{6} + \frac {D a^{2} b c^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 A a^{2} b d^{2}}{5} + \frac {3 A a b^{2} c^{2}}{5} + \frac {6 B a^{2} b c d}{5} + \frac {C a^{3} d^{2}}{5} + \frac {3 C a^{2} b c^{2}}{5} + \frac {2 D a^{3} c d}{5}\right ) + x^{4} \cdot \left (\frac {3 A a^{2} b c d}{2} + \frac {B a^{3} d^{2}}{4} + \frac {3 B a^{2} b c^{2}}{4} + \frac {C a^{3} c d}{2} + \frac {D a^{3} c^{2}}{4}\right ) + x^{3} \left (\frac {A a^{3} d^{2}}{3} + A a^{2} b c^{2} + \frac {2 B a^{3} c d}{3} + \frac {C a^{3} c^{2}}{3}\right ) + x^{2} \left (A a^{3} c d + \frac {B a^{3} c^{2}}{2}\right ) \] Input:

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

Output:

A*a**3*c**2*x + D*b**3*d**2*x**12/12 + x**11*(C*b**3*d**2/11 + 2*D*b**3*c* 
d/11) + x**10*(B*b**3*d**2/10 + C*b**3*c*d/5 + 3*D*a*b**2*d**2/10 + D*b**3 
*c**2/10) + x**9*(A*b**3*d**2/9 + 2*B*b**3*c*d/9 + C*a*b**2*d**2/3 + C*b** 
3*c**2/9 + 2*D*a*b**2*c*d/3) + x**8*(A*b**3*c*d/4 + 3*B*a*b**2*d**2/8 + B* 
b**3*c**2/8 + 3*C*a*b**2*c*d/4 + 3*D*a**2*b*d**2/8 + 3*D*a*b**2*c**2/8) + 
x**7*(3*A*a*b**2*d**2/7 + A*b**3*c**2/7 + 6*B*a*b**2*c*d/7 + 3*C*a**2*b*d* 
*2/7 + 3*C*a*b**2*c**2/7 + 6*D*a**2*b*c*d/7) + x**6*(A*a*b**2*c*d + B*a**2 
*b*d**2/2 + B*a*b**2*c**2/2 + C*a**2*b*c*d + D*a**3*d**2/6 + D*a**2*b*c**2 
/2) + x**5*(3*A*a**2*b*d**2/5 + 3*A*a*b**2*c**2/5 + 6*B*a**2*b*c*d/5 + C*a 
**3*d**2/5 + 3*C*a**2*b*c**2/5 + 2*D*a**3*c*d/5) + x**4*(3*A*a**2*b*c*d/2 
+ B*a**3*d**2/4 + 3*B*a**2*b*c**2/4 + C*a**3*c*d/2 + D*a**3*c**2/4) + x**3 
*(A*a**3*d**2/3 + A*a**2*b*c**2 + 2*B*a**3*c*d/3 + C*a**3*c**2/3) + x**2*( 
A*a**3*c*d + B*a**3*c**2/2)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.13 \[ \int (c+d x)^2 \left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{12} \, D b^{3} d^{2} x^{12} + \frac {1}{11} \, {\left (2 \, D b^{3} c d + C b^{3} d^{2}\right )} x^{11} + \frac {1}{10} \, {\left (D b^{3} c^{2} + 2 \, C b^{3} c d + {\left (3 \, D a b^{2} + B b^{3}\right )} d^{2}\right )} x^{10} + \frac {1}{9} \, {\left (C b^{3} c^{2} + 2 \, {\left (3 \, D a b^{2} + B b^{3}\right )} c d + {\left (3 \, C a b^{2} + A b^{3}\right )} d^{2}\right )} x^{9} + \frac {1}{8} \, {\left ({\left (3 \, D a b^{2} + B b^{3}\right )} c^{2} + 2 \, {\left (3 \, C a b^{2} + A b^{3}\right )} c d + 3 \, {\left (D a^{2} b + B a b^{2}\right )} d^{2}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, C a b^{2} + A b^{3}\right )} c^{2} + 6 \, {\left (D a^{2} b + B a b^{2}\right )} c d + 3 \, {\left (C a^{2} b + A a b^{2}\right )} d^{2}\right )} x^{7} + A a^{3} c^{2} x + \frac {1}{6} \, {\left (3 \, {\left (D a^{2} b + B a b^{2}\right )} c^{2} + 6 \, {\left (C a^{2} b + A a b^{2}\right )} c d + {\left (D a^{3} + 3 \, B a^{2} b\right )} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, {\left (C a^{2} b + A a b^{2}\right )} c^{2} + 2 \, {\left (D a^{3} + 3 \, B a^{2} b\right )} c d + {\left (C a^{3} + 3 \, A a^{2} b\right )} d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{3} d^{2} + {\left (D a^{3} + 3 \, B a^{2} b\right )} c^{2} + 2 \, {\left (C a^{3} + 3 \, A a^{2} b\right )} c d\right )} x^{4} + \frac {1}{3} \, {\left (2 \, B a^{3} c d + A a^{3} d^{2} + {\left (C a^{3} + 3 \, A a^{2} b\right )} c^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} c^{2} + 2 \, A a^{3} c d\right )} x^{2} \] Input:

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

Output:

1/12*D*b^3*d^2*x^12 + 1/11*(2*D*b^3*c*d + C*b^3*d^2)*x^11 + 1/10*(D*b^3*c^ 
2 + 2*C*b^3*c*d + (3*D*a*b^2 + B*b^3)*d^2)*x^10 + 1/9*(C*b^3*c^2 + 2*(3*D* 
a*b^2 + B*b^3)*c*d + (3*C*a*b^2 + A*b^3)*d^2)*x^9 + 1/8*((3*D*a*b^2 + B*b^ 
3)*c^2 + 2*(3*C*a*b^2 + A*b^3)*c*d + 3*(D*a^2*b + B*a*b^2)*d^2)*x^8 + 1/7* 
((3*C*a*b^2 + A*b^3)*c^2 + 6*(D*a^2*b + B*a*b^2)*c*d + 3*(C*a^2*b + A*a*b^ 
2)*d^2)*x^7 + A*a^3*c^2*x + 1/6*(3*(D*a^2*b + B*a*b^2)*c^2 + 6*(C*a^2*b + 
A*a*b^2)*c*d + (D*a^3 + 3*B*a^2*b)*d^2)*x^6 + 1/5*(3*(C*a^2*b + A*a*b^2)*c 
^2 + 2*(D*a^3 + 3*B*a^2*b)*c*d + (C*a^3 + 3*A*a^2*b)*d^2)*x^5 + 1/4*(B*a^3 
*d^2 + (D*a^3 + 3*B*a^2*b)*c^2 + 2*(C*a^3 + 3*A*a^2*b)*c*d)*x^4 + 1/3*(2*B 
*a^3*c*d + A*a^3*d^2 + (C*a^3 + 3*A*a^2*b)*c^2)*x^3 + 1/2*(B*a^3*c^2 + 2*A 
*a^3*c*d)*x^2
 

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.38 \[ \int (c+d x)^2 \left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{12} \, D b^{3} d^{2} x^{12} + \frac {2}{11} \, D b^{3} c d x^{11} + \frac {1}{11} \, C b^{3} d^{2} x^{11} + \frac {1}{10} \, D b^{3} c^{2} x^{10} + \frac {1}{5} \, C b^{3} c d x^{10} + \frac {3}{10} \, D a b^{2} d^{2} x^{10} + \frac {1}{10} \, B b^{3} d^{2} x^{10} + \frac {1}{9} \, C b^{3} c^{2} x^{9} + \frac {2}{3} \, D a b^{2} c d x^{9} + \frac {2}{9} \, B b^{3} c d x^{9} + \frac {1}{3} \, C a b^{2} d^{2} x^{9} + \frac {1}{9} \, A b^{3} d^{2} x^{9} + \frac {3}{8} \, D a b^{2} c^{2} x^{8} + \frac {1}{8} \, B b^{3} c^{2} x^{8} + \frac {3}{4} \, C a b^{2} c d x^{8} + \frac {1}{4} \, A b^{3} c d x^{8} + \frac {3}{8} \, D a^{2} b d^{2} x^{8} + \frac {3}{8} \, B a b^{2} d^{2} x^{8} + \frac {3}{7} \, C a b^{2} c^{2} x^{7} + \frac {1}{7} \, A b^{3} c^{2} x^{7} + \frac {6}{7} \, D a^{2} b c d x^{7} + \frac {6}{7} \, B a b^{2} c d x^{7} + \frac {3}{7} \, C a^{2} b d^{2} x^{7} + \frac {3}{7} \, A a b^{2} d^{2} x^{7} + \frac {1}{2} \, D a^{2} b c^{2} x^{6} + \frac {1}{2} \, B a b^{2} c^{2} x^{6} + C a^{2} b c d x^{6} + A a b^{2} c d x^{6} + \frac {1}{6} \, D a^{3} d^{2} x^{6} + \frac {1}{2} \, B a^{2} b d^{2} x^{6} + \frac {3}{5} \, C a^{2} b c^{2} x^{5} + \frac {3}{5} \, A a b^{2} c^{2} x^{5} + \frac {2}{5} \, D a^{3} c d x^{5} + \frac {6}{5} \, B a^{2} b c d x^{5} + \frac {1}{5} \, C a^{3} d^{2} x^{5} + \frac {3}{5} \, A a^{2} b d^{2} x^{5} + \frac {1}{4} \, D a^{3} c^{2} x^{4} + \frac {3}{4} \, B a^{2} b c^{2} x^{4} + \frac {1}{2} \, C a^{3} c d x^{4} + \frac {3}{2} \, A a^{2} b c d x^{4} + \frac {1}{4} \, B a^{3} d^{2} x^{4} + \frac {1}{3} \, C a^{3} c^{2} x^{3} + A a^{2} b c^{2} x^{3} + \frac {2}{3} \, B a^{3} c d x^{3} + \frac {1}{3} \, A a^{3} d^{2} x^{3} + \frac {1}{2} \, B a^{3} c^{2} x^{2} + A a^{3} c d x^{2} + A a^{3} c^{2} x \] Input:

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

Output:

1/12*D*b^3*d^2*x^12 + 2/11*D*b^3*c*d*x^11 + 1/11*C*b^3*d^2*x^11 + 1/10*D*b 
^3*c^2*x^10 + 1/5*C*b^3*c*d*x^10 + 3/10*D*a*b^2*d^2*x^10 + 1/10*B*b^3*d^2* 
x^10 + 1/9*C*b^3*c^2*x^9 + 2/3*D*a*b^2*c*d*x^9 + 2/9*B*b^3*c*d*x^9 + 1/3*C 
*a*b^2*d^2*x^9 + 1/9*A*b^3*d^2*x^9 + 3/8*D*a*b^2*c^2*x^8 + 1/8*B*b^3*c^2*x 
^8 + 3/4*C*a*b^2*c*d*x^8 + 1/4*A*b^3*c*d*x^8 + 3/8*D*a^2*b*d^2*x^8 + 3/8*B 
*a*b^2*d^2*x^8 + 3/7*C*a*b^2*c^2*x^7 + 1/7*A*b^3*c^2*x^7 + 6/7*D*a^2*b*c*d 
*x^7 + 6/7*B*a*b^2*c*d*x^7 + 3/7*C*a^2*b*d^2*x^7 + 3/7*A*a*b^2*d^2*x^7 + 1 
/2*D*a^2*b*c^2*x^6 + 1/2*B*a*b^2*c^2*x^6 + C*a^2*b*c*d*x^6 + A*a*b^2*c*d*x 
^6 + 1/6*D*a^3*d^2*x^6 + 1/2*B*a^2*b*d^2*x^6 + 3/5*C*a^2*b*c^2*x^5 + 3/5*A 
*a*b^2*c^2*x^5 + 2/5*D*a^3*c*d*x^5 + 6/5*B*a^2*b*c*d*x^5 + 1/5*C*a^3*d^2*x 
^5 + 3/5*A*a^2*b*d^2*x^5 + 1/4*D*a^3*c^2*x^4 + 3/4*B*a^2*b*c^2*x^4 + 1/2*C 
*a^3*c*d*x^4 + 3/2*A*a^2*b*c*d*x^4 + 1/4*B*a^3*d^2*x^4 + 1/3*C*a^3*c^2*x^3 
 + A*a^2*b*c^2*x^3 + 2/3*B*a^3*c*d*x^3 + 1/3*A*a^3*d^2*x^3 + 1/2*B*a^3*c^2 
*x^2 + A*a^3*c*d*x^2 + A*a^3*c^2*x
 

Mupad [B] (verification not implemented)

Time = 24.23 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.38 \[ \int (c+d x)^2 \left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {a^3\,c^2\,x^4\,D}{4}+\frac {a^3\,d^2\,x^6\,D}{6}+\frac {b^3\,c^2\,x^{10}\,D}{10}+\frac {b^3\,d^2\,x^{12}\,D}{12}+A\,a^3\,c^2\,x+\frac {B\,a^3\,c^2\,x^2}{2}+\frac {A\,a^3\,d^2\,x^3}{3}+\frac {A\,b^3\,c^2\,x^7}{7}+\frac {C\,a^3\,c^2\,x^3}{3}+\frac {B\,a^3\,d^2\,x^4}{4}+\frac {B\,b^3\,c^2\,x^8}{8}+\frac {A\,b^3\,d^2\,x^9}{9}+\frac {C\,a^3\,d^2\,x^5}{5}+\frac {C\,b^3\,c^2\,x^9}{9}+\frac {B\,b^3\,d^2\,x^{10}}{10}+\frac {C\,b^3\,d^2\,x^{11}}{11}+\frac {2\,B\,a^3\,c\,d\,x^3}{3}+\frac {A\,b^3\,c\,d\,x^8}{4}+\frac {C\,a^3\,c\,d\,x^4}{2}+\frac {2\,B\,b^3\,c\,d\,x^9}{9}+\frac {C\,b^3\,c\,d\,x^{10}}{5}+\frac {2\,a^3\,c\,d\,x^5\,D}{5}+\frac {2\,b^3\,c\,d\,x^{11}\,D}{11}+A\,a^2\,b\,c^2\,x^3+\frac {3\,A\,a\,b^2\,c^2\,x^5}{5}+\frac {3\,B\,a^2\,b\,c^2\,x^4}{4}+\frac {3\,A\,a^2\,b\,d^2\,x^5}{5}+\frac {B\,a\,b^2\,c^2\,x^6}{2}+\frac {3\,A\,a\,b^2\,d^2\,x^7}{7}+\frac {3\,C\,a^2\,b\,c^2\,x^5}{5}+\frac {B\,a^2\,b\,d^2\,x^6}{2}+\frac {3\,C\,a\,b^2\,c^2\,x^7}{7}+\frac {3\,B\,a\,b^2\,d^2\,x^8}{8}+\frac {3\,C\,a^2\,b\,d^2\,x^7}{7}+\frac {C\,a\,b^2\,d^2\,x^9}{3}+\frac {a^2\,b\,c^2\,x^6\,D}{2}+\frac {3\,a\,b^2\,c^2\,x^8\,D}{8}+\frac {3\,a^2\,b\,d^2\,x^8\,D}{8}+\frac {3\,a\,b^2\,d^2\,x^{10}\,D}{10}+A\,a^3\,c\,d\,x^2+\frac {6\,a^2\,b\,c\,d\,x^7\,D}{7}+\frac {2\,a\,b^2\,c\,d\,x^9\,D}{3}+\frac {3\,A\,a^2\,b\,c\,d\,x^4}{2}+A\,a\,b^2\,c\,d\,x^6+\frac {6\,B\,a^2\,b\,c\,d\,x^5}{5}+\frac {6\,B\,a\,b^2\,c\,d\,x^7}{7}+C\,a^2\,b\,c\,d\,x^6+\frac {3\,C\,a\,b^2\,c\,d\,x^8}{4} \] Input:

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

Output:

(a^3*c^2*x^4*D)/4 + (a^3*d^2*x^6*D)/6 + (b^3*c^2*x^10*D)/10 + (b^3*d^2*x^1 
2*D)/12 + A*a^3*c^2*x + (B*a^3*c^2*x^2)/2 + (A*a^3*d^2*x^3)/3 + (A*b^3*c^2 
*x^7)/7 + (C*a^3*c^2*x^3)/3 + (B*a^3*d^2*x^4)/4 + (B*b^3*c^2*x^8)/8 + (A*b 
^3*d^2*x^9)/9 + (C*a^3*d^2*x^5)/5 + (C*b^3*c^2*x^9)/9 + (B*b^3*d^2*x^10)/1 
0 + (C*b^3*d^2*x^11)/11 + (2*B*a^3*c*d*x^3)/3 + (A*b^3*c*d*x^8)/4 + (C*a^3 
*c*d*x^4)/2 + (2*B*b^3*c*d*x^9)/9 + (C*b^3*c*d*x^10)/5 + (2*a^3*c*d*x^5*D) 
/5 + (2*b^3*c*d*x^11*D)/11 + A*a^2*b*c^2*x^3 + (3*A*a*b^2*c^2*x^5)/5 + (3* 
B*a^2*b*c^2*x^4)/4 + (3*A*a^2*b*d^2*x^5)/5 + (B*a*b^2*c^2*x^6)/2 + (3*A*a* 
b^2*d^2*x^7)/7 + (3*C*a^2*b*c^2*x^5)/5 + (B*a^2*b*d^2*x^6)/2 + (3*C*a*b^2* 
c^2*x^7)/7 + (3*B*a*b^2*d^2*x^8)/8 + (3*C*a^2*b*d^2*x^7)/7 + (C*a*b^2*d^2* 
x^9)/3 + (a^2*b*c^2*x^6*D)/2 + (3*a*b^2*c^2*x^8*D)/8 + (3*a^2*b*d^2*x^8*D) 
/8 + (3*a*b^2*d^2*x^10*D)/10 + A*a^3*c*d*x^2 + (6*a^2*b*c*d*x^7*D)/7 + (2* 
a*b^2*c*d*x^9*D)/3 + (3*A*a^2*b*c*d*x^4)/2 + A*a*b^2*c*d*x^6 + (6*B*a^2*b* 
c*d*x^5)/5 + (6*B*a*b^2*c*d*x^7)/7 + C*a^2*b*c*d*x^6 + (3*C*a*b^2*c*d*x^8) 
/4
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.14 \[ \int (c+d x)^2 \left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {x \left (2310 b^{3} d^{3} x^{11}+7560 b^{3} c \,d^{2} x^{10}+8316 a \,b^{2} d^{3} x^{9}+2772 b^{4} d^{2} x^{9}+8316 b^{3} c^{2} d \,x^{9}+3080 a \,b^{3} d^{2} x^{8}+27720 a \,b^{2} c \,d^{2} x^{8}+6160 b^{4} c d \,x^{8}+3080 b^{3} c^{3} x^{8}+10395 a^{2} b \,d^{3} x^{7}+6930 a \,b^{3} c d \,x^{7}+10395 a \,b^{3} d^{2} x^{7}+31185 a \,b^{2} c^{2} d \,x^{7}+3465 b^{4} c^{2} x^{7}+11880 a^{2} b^{2} d^{2} x^{6}+35640 a^{2} b c \,d^{2} x^{6}+3960 a \,b^{3} c^{2} x^{6}+23760 a \,b^{3} c d \,x^{6}+11880 a \,b^{2} c^{3} x^{6}+4620 a^{3} d^{3} x^{5}+27720 a^{2} b^{2} c d \,x^{5}+13860 a^{2} b^{2} d^{2} x^{5}+41580 a^{2} b \,c^{2} d \,x^{5}+13860 a \,b^{3} c^{2} x^{5}+16632 a^{3} b \,d^{2} x^{4}+16632 a^{3} c \,d^{2} x^{4}+16632 a^{2} b^{2} c^{2} x^{4}+33264 a^{2} b^{2} c d \,x^{4}+16632 a^{2} b \,c^{3} x^{4}+41580 a^{3} b c d \,x^{3}+6930 a^{3} b \,d^{2} x^{3}+20790 a^{3} c^{2} d \,x^{3}+20790 a^{2} b^{2} c^{2} x^{3}+9240 a^{4} d^{2} x^{2}+27720 a^{3} b \,c^{2} x^{2}+18480 a^{3} b c d \,x^{2}+9240 a^{3} c^{3} x^{2}+27720 a^{4} c d x +13860 a^{3} b \,c^{2} x +27720 a^{4} c^{2}\right )}{27720} \] Input:

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

Output:

(x*(27720*a**4*c**2 + 27720*a**4*c*d*x + 9240*a**4*d**2*x**2 + 27720*a**3* 
b*c**2*x**2 + 13860*a**3*b*c**2*x + 41580*a**3*b*c*d*x**3 + 18480*a**3*b*c 
*d*x**2 + 16632*a**3*b*d**2*x**4 + 6930*a**3*b*d**2*x**3 + 9240*a**3*c**3* 
x**2 + 20790*a**3*c**2*d*x**3 + 16632*a**3*c*d**2*x**4 + 4620*a**3*d**3*x* 
*5 + 16632*a**2*b**2*c**2*x**4 + 20790*a**2*b**2*c**2*x**3 + 27720*a**2*b* 
*2*c*d*x**5 + 33264*a**2*b**2*c*d*x**4 + 11880*a**2*b**2*d**2*x**6 + 13860 
*a**2*b**2*d**2*x**5 + 16632*a**2*b*c**3*x**4 + 41580*a**2*b*c**2*d*x**5 + 
 35640*a**2*b*c*d**2*x**6 + 10395*a**2*b*d**3*x**7 + 3960*a*b**3*c**2*x**6 
 + 13860*a*b**3*c**2*x**5 + 6930*a*b**3*c*d*x**7 + 23760*a*b**3*c*d*x**6 + 
 3080*a*b**3*d**2*x**8 + 10395*a*b**3*d**2*x**7 + 11880*a*b**2*c**3*x**6 + 
 31185*a*b**2*c**2*d*x**7 + 27720*a*b**2*c*d**2*x**8 + 8316*a*b**2*d**3*x* 
*9 + 3465*b**4*c**2*x**7 + 6160*b**4*c*d*x**8 + 2772*b**4*d**2*x**9 + 3080 
*b**3*c**3*x**8 + 8316*b**3*c**2*d*x**9 + 7560*b**3*c*d**2*x**10 + 2310*b* 
*3*d**3*x**11))/27720