Integrand size = 32, antiderivative size = 284 \[ \int (c+d x)^2 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=a^2 A c^2 x+\frac {1}{3} a \left (a c (c C+2 B d)+A \left (2 b c^2+a d^2\right )\right ) x^3+\frac {1}{4} a^2 \left (2 c C d+B d^2+c^2 D\right ) x^4+\frac {1}{5} \left (A b \left (b c^2+2 a d^2\right )+a (2 b c (c C+2 B d)+a d (C d+2 c D))\right ) x^5+\frac {1}{6} a \left (a d^2 D+2 b \left (2 c C d+B d^2+c^2 D\right )\right ) x^6+\frac {1}{7} b \left (b \left (c^2 C+2 B c d+A d^2\right )+2 a d (C d+2 c D)\right ) x^7+\frac {1}{8} b \left (2 a d^2 D+b \left (2 c C d+B d^2+c^2 D\right )\right ) x^8+\frac {1}{9} b^2 d (C d+2 c D) x^9+\frac {1}{10} b^2 d^2 D x^{10}+\frac {c (B c+2 A d) \left (a+b x^2\right )^3}{6 b} \] Output:
a^2*A*c^2*x+1/3*a*(a*c*(2*B*d+C*c)+A*(a*d^2+2*b*c^2))*x^3+1/4*a^2*(B*d^2+2 *C*c*d+D*c^2)*x^4+1/5*(A*b*(2*a*d^2+b*c^2)+a*(2*b*c*(2*B*d+C*c)+a*d*(C*d+2 *D*c)))*x^5+1/6*a*(a*d^2*D+2*b*(B*d^2+2*C*c*d+D*c^2))*x^6+1/7*b*(b*(A*d^2+ 2*B*c*d+C*c^2)+2*a*d*(C*d+2*D*c))*x^7+1/8*b*(2*a*d^2*D+b*(B*d^2+2*C*c*d+D* c^2))*x^8+1/9*b^2*d*(C*d+2*D*c)*x^9+1/10*b^2*d^2*D*x^10+1/6*c*(2*A*d+B*c)* (b*x^2+a)^3/b
Time = 0.17 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.08 \[ \int (c+d x)^2 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=a^2 A c^2 x+\frac {1}{2} a^2 c (B c+2 A d) x^2+\frac {1}{3} a \left (a c (c C+2 B d)+A \left (2 b c^2+a d^2\right )\right ) x^3+\frac {1}{4} a \left (2 b c (B c+2 A d)+a \left (2 c C d+B d^2+c^2 D\right )\right ) x^4+\frac {1}{5} \left (A b \left (b c^2+2 a d^2\right )+a (2 b c (c C+2 B d)+a d (C d+2 c D))\right ) x^5+\frac {1}{6} \left (b^2 c (B c+2 A d)+a^2 d^2 D+2 a b \left (2 c C d+B d^2+c^2 D\right )\right ) x^6+\frac {1}{7} b \left (b \left (c^2 C+2 B c d+A d^2\right )+2 a d (C d+2 c D)\right ) x^7+\frac {1}{8} b \left (2 a d^2 D+b \left (2 c C d+B d^2+c^2 D\right )\right ) x^8+\frac {1}{9} b^2 d (C d+2 c D) x^9+\frac {1}{10} b^2 d^2 D x^{10} \] Input:
Integrate[(c + d*x)^2*(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]
Output:
a^2*A*c^2*x + (a^2*c*(B*c + 2*A*d)*x^2)/2 + (a*(a*c*(c*C + 2*B*d) + A*(2*b *c^2 + a*d^2))*x^3)/3 + (a*(2*b*c*(B*c + 2*A*d) + a*(2*c*C*d + B*d^2 + c^2 *D))*x^4)/4 + ((A*b*(b*c^2 + 2*a*d^2) + a*(2*b*c*(c*C + 2*B*d) + a*d*(C*d + 2*c*D)))*x^5)/5 + ((b^2*c*(B*c + 2*A*d) + a^2*d^2*D + 2*a*b*(2*c*C*d + B *d^2 + c^2*D))*x^6)/6 + (b*(b*(c^2*C + 2*B*c*d + A*d^2) + 2*a*d*(C*d + 2*c *D))*x^7)/7 + (b*(2*a*d^2*D + b*(2*c*C*d + B*d^2 + c^2*D))*x^8)/8 + (b^2*d *(C*d + 2*c*D)*x^9)/9 + (b^2*d^2*D*x^10)/10
Time = 0.74 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, 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.
| \(\displaystyle \int \left (a+b x^2\right )^2 (c+d x)^2 \left (A+B x+C x^2+D x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2017 |
| \(\displaystyle \int \left (b x^2+a\right )^2 \left ((c+d x)^2 \left (D x^3+C x^2+B x+A\right )-\left (B c^2+2 A d c\right ) x\right )dx+\frac {c \left (a+b x^2\right )^3 (2 A d+B c)}{6 b}\) |
| \(\Big \downarrow \) 2341 |
| \(\displaystyle \int \left (b^2 d^2 D x^9+b^2 d (C d+2 c D) x^8+b \left (2 a D d^2+b \left (D c^2+2 C d c+B d^2\right )\right ) x^7+b \left (b \left (C c^2+2 B d c+A d^2\right )+2 a d (C d+2 c D)\right ) x^6+a \left (a D d^2+2 b \left (D c^2+2 C d c+B d^2\right )\right ) x^5+\left (A b \left (b c^2+2 a d^2\right )+a (2 b c (c C+2 B d)+a d (C d+2 c D))\right ) x^4+a^2 \left (D c^2+2 C d c+B d^2\right ) x^3+a \left (a c (c C+2 B d)+A \left (2 b c^2+a d^2\right )\right ) x^2+a^2 A c^2\right )dx+\frac {c \left (a+b x^2\right )^3 (2 A d+B c)}{6 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 A c^2 x+\frac {1}{4} a^2 x^4 \left (B d^2+c^2 D+2 c C d\right )+\frac {1}{7} b x^7 \left (2 a d (2 c D+C d)+b \left (A d^2+2 B c d+c^2 C\right )\right )+\frac {1}{5} x^5 \left (A b \left (2 a d^2+b c^2\right )+a (a d (2 c D+C d)+2 b c (2 B d+c C))\right )+\frac {1}{3} a x^3 \left (A \left (a d^2+2 b c^2\right )+a c (2 B d+c C)\right )+\frac {c \left (a+b x^2\right )^3 (2 A d+B c)}{6 b}+\frac {1}{8} b x^8 \left (2 a d^2 D+b \left (B d^2+c^2 D+2 c C d\right )\right )+\frac {1}{6} a x^6 \left (a d^2 D+2 b \left (B d^2+c^2 D+2 c C d\right )\right )+\frac {1}{9} b^2 d x^9 (2 c D+C d)+\frac {1}{10} b^2 d^2 D x^{10}\) |
Input:
Int[(c + d*x)^2*(a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3),x]
Output:
a^2*A*c^2*x + (a*(a*c*(c*C + 2*B*d) + A*(2*b*c^2 + a*d^2))*x^3)/3 + (a^2*( 2*c*C*d + B*d^2 + c^2*D)*x^4)/4 + ((A*b*(b*c^2 + 2*a*d^2) + a*(2*b*c*(c*C + 2*B*d) + a*d*(C*d + 2*c*D)))*x^5)/5 + (a*(a*d^2*D + 2*b*(2*c*C*d + B*d^2 + c^2*D))*x^6)/6 + (b*(b*(c^2*C + 2*B*c*d + A*d^2) + 2*a*d*(C*d + 2*c*D)) *x^7)/7 + (b*(2*a*d^2*D + b*(2*c*C*d + B*d^2 + c^2*D))*x^8)/8 + (b^2*d*(C* d + 2*c*D)*x^9)/9 + (b^2*d^2*D*x^10)/10 + (c*(B*c + 2*A*d)*(a + b*x^2)^3)/ (6*b)
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]]
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]
Time = 0.74 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.21
| method | result | size |
| norman | \(\frac {b^{2} d^{2} D x^{10}}{10}+\left (\frac {1}{9} b^{2} d^{2} C +\frac {2}{9} c d \,b^{2} D\right ) x^{9}+\left (\frac {1}{8} B \,b^{2} d^{2}+\frac {1}{4} c d \,b^{2} C +\frac {1}{4} D a b \,d^{2}+\frac {1}{8} D b^{2} c^{2}\right ) x^{8}+\left (\frac {1}{7} A \,b^{2} d^{2}+\frac {2}{7} B \,b^{2} c d +\frac {2}{7} C a b \,d^{2}+\frac {1}{7} C \,b^{2} c^{2}+\frac {4}{7} a b c d D\right ) x^{7}+\left (\frac {1}{3} A \,b^{2} c d +\frac {1}{3} B a b \,d^{2}+\frac {1}{6} B \,b^{2} c^{2}+\frac {2}{3} a b c d C +\frac {1}{6} a^{2} d^{2} D+\frac {1}{3} D a b \,c^{2}\right ) x^{6}+\left (\frac {2}{5} A a b \,d^{2}+\frac {1}{5} A \,b^{2} c^{2}+\frac {4}{5} a b B c d +\frac {1}{5} a^{2} C \,d^{2}+\frac {2}{5} C a b \,c^{2}+\frac {2}{5} a^{2} c d D\right ) x^{5}+\left (A a b c d +\frac {1}{4} a^{2} B \,d^{2}+\frac {1}{2} B a b \,c^{2}+\frac {1}{2} C \,a^{2} c d +\frac {1}{4} a^{2} c^{2} D\right ) x^{4}+\left (\frac {1}{3} A \,d^{2} a^{2}+\frac {2}{3} A a b \,c^{2}+\frac {2}{3} B \,a^{2} c d +\frac {1}{3} C \,a^{2} c^{2}\right ) x^{3}+\left (A \,a^{2} c d +\frac {1}{2} B \,a^{2} c^{2}\right ) x^{2}+a^{2} A \,c^{2} x\) | \(343\) |
| default | \(\frac {b^{2} d^{2} D x^{10}}{10}+\frac {\left (b^{2} d^{2} C +2 c d \,b^{2} D\right ) x^{9}}{9}+\frac {\left (\left (2 a b \,d^{2}+b^{2} c^{2}\right ) D+2 c d \,b^{2} C +B \,b^{2} d^{2}\right ) x^{8}}{8}+\frac {\left (4 a b c d D+\left (2 a b \,d^{2}+b^{2} c^{2}\right ) C +2 B \,b^{2} c d +A \,b^{2} d^{2}\right ) x^{7}}{7}+\frac {\left (\left (a^{2} d^{2}+2 a b \,c^{2}\right ) D+4 a b c d C +\left (2 a b \,d^{2}+b^{2} c^{2}\right ) B +2 A \,b^{2} c d \right ) x^{6}}{6}+\frac {\left (2 a^{2} c d D+\left (a^{2} d^{2}+2 a b \,c^{2}\right ) C +4 a b B c d +\left (2 a b \,d^{2}+b^{2} c^{2}\right ) A \right ) x^{5}}{5}+\frac {\left (a^{2} c^{2} D+2 C \,a^{2} c d +\left (a^{2} d^{2}+2 a b \,c^{2}\right ) B +4 A a b c d \right ) x^{4}}{4}+\frac {\left (C \,a^{2} c^{2}+2 B \,a^{2} c d +\left (a^{2} d^{2}+2 a b \,c^{2}\right ) A \right ) x^{3}}{3}+\frac {\left (2 A \,a^{2} c d +B \,a^{2} c^{2}\right ) x^{2}}{2}+a^{2} A \,c^{2} x\) | \(347\) |
| gosper | \(\frac {1}{4} x^{8} c d \,b^{2} C +\frac {1}{3} x^{6} A \,b^{2} c d +\frac {2}{3} x^{3} B \,a^{2} c d +x^{2} A \,a^{2} c d +\frac {1}{3} x^{3} A \,d^{2} a^{2}+\frac {1}{3} x^{3} C \,a^{2} c^{2}+\frac {1}{3} x^{6} D a b \,c^{2}+\frac {2}{5} x^{5} A a b \,d^{2}+\frac {1}{2} x^{4} C \,a^{2} c d +\frac {1}{2} x^{2} B \,a^{2} c^{2}+\frac {2}{7} x^{7} C a b \,d^{2}+\frac {1}{3} x^{6} B a b \,d^{2}+\frac {2}{9} x^{9} c d \,b^{2} D+\frac {2}{7} x^{7} B \,b^{2} c d +\frac {1}{7} x^{7} C \,b^{2} c^{2}+\frac {4}{5} x^{5} a b B c d +x^{4} A a b c d +\frac {2}{5} x^{5} C a b \,c^{2}+\frac {2}{5} x^{5} a^{2} c d D+\frac {1}{4} x^{8} D a b \,d^{2}+\frac {1}{6} x^{6} a^{2} d^{2} D+\frac {1}{5} x^{5} A \,b^{2} c^{2}+\frac {1}{5} x^{5} a^{2} C \,d^{2}+\frac {1}{4} x^{4} a^{2} B \,d^{2}+\frac {1}{4} x^{4} a^{2} c^{2} D+\frac {1}{9} x^{9} b^{2} d^{2} C +\frac {2}{3} a A b \,c^{2} x^{3}+\frac {4}{7} x^{7} a b c d D+\frac {1}{6} B \,b^{2} c^{2} x^{6}+\frac {1}{8} x^{8} D b^{2} c^{2}+\frac {1}{7} x^{7} A \,b^{2} d^{2}+\frac {2}{3} x^{6} a b c d C +\frac {1}{8} x^{8} B \,b^{2} d^{2}+\frac {1}{10} b^{2} d^{2} D x^{10}+a^{2} A \,c^{2} x +\frac {1}{2} B a b \,c^{2} x^{4}\) | \(405\) |
| parallelrisch | \(\frac {1}{4} x^{8} c d \,b^{2} C +\frac {1}{3} x^{6} A \,b^{2} c d +\frac {2}{3} x^{3} B \,a^{2} c d +x^{2} A \,a^{2} c d +\frac {1}{3} x^{3} A \,d^{2} a^{2}+\frac {1}{3} x^{3} C \,a^{2} c^{2}+\frac {1}{3} x^{6} D a b \,c^{2}+\frac {2}{5} x^{5} A a b \,d^{2}+\frac {1}{2} x^{4} C \,a^{2} c d +\frac {1}{2} x^{2} B \,a^{2} c^{2}+\frac {2}{7} x^{7} C a b \,d^{2}+\frac {1}{3} x^{6} B a b \,d^{2}+\frac {2}{9} x^{9} c d \,b^{2} D+\frac {2}{7} x^{7} B \,b^{2} c d +\frac {1}{7} x^{7} C \,b^{2} c^{2}+\frac {4}{5} x^{5} a b B c d +x^{4} A a b c d +\frac {2}{5} x^{5} C a b \,c^{2}+\frac {2}{5} x^{5} a^{2} c d D+\frac {1}{4} x^{8} D a b \,d^{2}+\frac {1}{6} x^{6} a^{2} d^{2} D+\frac {1}{5} x^{5} A \,b^{2} c^{2}+\frac {1}{5} x^{5} a^{2} C \,d^{2}+\frac {1}{4} x^{4} a^{2} B \,d^{2}+\frac {1}{4} x^{4} a^{2} c^{2} D+\frac {1}{9} x^{9} b^{2} d^{2} C +\frac {2}{3} a A b \,c^{2} x^{3}+\frac {4}{7} x^{7} a b c d D+\frac {1}{6} B \,b^{2} c^{2} x^{6}+\frac {1}{8} x^{8} D b^{2} c^{2}+\frac {1}{7} x^{7} A \,b^{2} d^{2}+\frac {2}{3} x^{6} a b c d C +\frac {1}{8} x^{8} B \,b^{2} d^{2}+\frac {1}{10} b^{2} d^{2} D x^{10}+a^{2} A \,c^{2} x +\frac {1}{2} B a b \,c^{2} x^{4}\) | \(405\) |
| orering | \(\frac {x \left (252 b^{2} d^{2} D x^{9}+280 C \,b^{2} d^{2} x^{8}+560 D b^{2} c d \,x^{8}+315 B \,b^{2} d^{2} x^{7}+630 C \,b^{2} c d \,x^{7}+630 D a b \,d^{2} x^{7}+315 D b^{2} c^{2} x^{7}+360 A \,b^{2} d^{2} x^{6}+720 B \,b^{2} c d \,x^{6}+720 C a b \,d^{2} x^{6}+360 C \,b^{2} c^{2} x^{6}+1440 D a b c d \,x^{6}+840 A \,b^{2} c d \,x^{5}+840 B a b \,d^{2} x^{5}+420 B \,b^{2} c^{2} x^{5}+1680 C a b c d \,x^{5}+420 D a^{2} d^{2} x^{5}+840 D a b \,c^{2} x^{5}+1008 A a b \,d^{2} x^{4}+504 A \,b^{2} c^{2} x^{4}+2016 B a b c d \,x^{4}+504 C \,a^{2} d^{2} x^{4}+1008 C a b \,c^{2} x^{4}+1008 D a^{2} c d \,x^{4}+2520 A a b c d \,x^{3}+630 B \,a^{2} d^{2} x^{3}+1260 B a b \,c^{2} x^{3}+1260 C \,a^{2} c d \,x^{3}+630 D a^{2} c^{2} x^{3}+840 A \,a^{2} d^{2} x^{2}+1680 A a b \,c^{2} x^{2}+1680 B \,a^{2} c d \,x^{2}+840 C \,a^{2} c^{2} x^{2}+2520 A \,a^{2} c d x +1260 B \,a^{2} c^{2} x +2520 a^{2} A \,c^{2}\right )}{2520}\) | \(406\) |
Input:
int((d*x+c)^2*(b*x^2+a)^2*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVERBOSE)
Output:
1/10*b^2*d^2*D*x^10+(1/9*b^2*d^2*C+2/9*c*d*b^2*D)*x^9+(1/8*B*b^2*d^2+1/4*c *d*b^2*C+1/4*D*a*b*d^2+1/8*D*b^2*c^2)*x^8+(1/7*A*b^2*d^2+2/7*B*b^2*c*d+2/7 *C*a*b*d^2+1/7*C*b^2*c^2+4/7*a*b*c*d*D)*x^7+(1/3*A*b^2*c*d+1/3*B*a*b*d^2+1 /6*B*b^2*c^2+2/3*a*b*c*d*C+1/6*a^2*d^2*D+1/3*D*a*b*c^2)*x^6+(2/5*A*a*b*d^2 +1/5*A*b^2*c^2+4/5*a*b*B*c*d+1/5*a^2*C*d^2+2/5*C*a*b*c^2+2/5*a^2*c*d*D)*x^ 5+(A*a*b*c*d+1/4*a^2*B*d^2+1/2*B*a*b*c^2+1/2*C*a^2*c*d+1/4*a^2*c^2*D)*x^4+ (1/3*A*d^2*a^2+2/3*A*a*b*c^2+2/3*B*a^2*c*d+1/3*C*a^2*c^2)*x^3+(A*a^2*c*d+1 /2*B*a^2*c^2)*x^2+a^2*A*c^2*x
Time = 0.06 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.16 \[ \int (c+d x)^2 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{10} \, D b^{2} d^{2} x^{10} + \frac {1}{9} \, {\left (2 \, D b^{2} c d + C b^{2} d^{2}\right )} x^{9} + \frac {1}{8} \, {\left (D b^{2} c^{2} + 2 \, C b^{2} c d + {\left (2 \, D a b + B b^{2}\right )} d^{2}\right )} x^{8} + \frac {1}{7} \, {\left (C b^{2} c^{2} + 2 \, {\left (2 \, D a b + B b^{2}\right )} c d + {\left (2 \, C a b + A b^{2}\right )} d^{2}\right )} x^{7} + \frac {1}{6} \, {\left ({\left (2 \, D a b + B b^{2}\right )} c^{2} + 2 \, {\left (2 \, C a b + A b^{2}\right )} c d + {\left (D a^{2} + 2 \, B a b\right )} d^{2}\right )} x^{6} + A a^{2} c^{2} x + \frac {1}{5} \, {\left ({\left (2 \, C a b + A b^{2}\right )} c^{2} + 2 \, {\left (D a^{2} + 2 \, B a b\right )} c d + {\left (C a^{2} + 2 \, A a b\right )} d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{2} d^{2} + {\left (D a^{2} + 2 \, B a b\right )} c^{2} + 2 \, {\left (C a^{2} + 2 \, A a b\right )} c d\right )} x^{4} + \frac {1}{3} \, {\left (2 \, B a^{2} c d + A a^{2} d^{2} + {\left (C a^{2} + 2 \, A a b\right )} c^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} c^{2} + 2 \, A a^{2} c d\right )} x^{2} \] Input:
integrate((d*x+c)^2*(b*x^2+a)^2*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")
Output:
1/10*D*b^2*d^2*x^10 + 1/9*(2*D*b^2*c*d + C*b^2*d^2)*x^9 + 1/8*(D*b^2*c^2 + 2*C*b^2*c*d + (2*D*a*b + B*b^2)*d^2)*x^8 + 1/7*(C*b^2*c^2 + 2*(2*D*a*b + B*b^2)*c*d + (2*C*a*b + A*b^2)*d^2)*x^7 + 1/6*((2*D*a*b + B*b^2)*c^2 + 2*( 2*C*a*b + A*b^2)*c*d + (D*a^2 + 2*B*a*b)*d^2)*x^6 + A*a^2*c^2*x + 1/5*((2* C*a*b + A*b^2)*c^2 + 2*(D*a^2 + 2*B*a*b)*c*d + (C*a^2 + 2*A*a*b)*d^2)*x^5 + 1/4*(B*a^2*d^2 + (D*a^2 + 2*B*a*b)*c^2 + 2*(C*a^2 + 2*A*a*b)*c*d)*x^4 + 1/3*(2*B*a^2*c*d + A*a^2*d^2 + (C*a^2 + 2*A*a*b)*c^2)*x^3 + 1/2*(B*a^2*c^2 + 2*A*a^2*c*d)*x^2
Time = 0.04 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.45 \[ \int (c+d x)^2 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=A a^{2} c^{2} x + \frac {D b^{2} d^{2} x^{10}}{10} + x^{9} \left (\frac {C b^{2} d^{2}}{9} + \frac {2 D b^{2} c d}{9}\right ) + x^{8} \left (\frac {B b^{2} d^{2}}{8} + \frac {C b^{2} c d}{4} + \frac {D a b d^{2}}{4} + \frac {D b^{2} c^{2}}{8}\right ) + x^{7} \left (\frac {A b^{2} d^{2}}{7} + \frac {2 B b^{2} c d}{7} + \frac {2 C a b d^{2}}{7} + \frac {C b^{2} c^{2}}{7} + \frac {4 D a b c d}{7}\right ) + x^{6} \left (\frac {A b^{2} c d}{3} + \frac {B a b d^{2}}{3} + \frac {B b^{2} c^{2}}{6} + \frac {2 C a b c d}{3} + \frac {D a^{2} d^{2}}{6} + \frac {D a b c^{2}}{3}\right ) + x^{5} \cdot \left (\frac {2 A a b d^{2}}{5} + \frac {A b^{2} c^{2}}{5} + \frac {4 B a b c d}{5} + \frac {C a^{2} d^{2}}{5} + \frac {2 C a b c^{2}}{5} + \frac {2 D a^{2} c d}{5}\right ) + x^{4} \left (A a b c d + \frac {B a^{2} d^{2}}{4} + \frac {B a b c^{2}}{2} + \frac {C a^{2} c d}{2} + \frac {D a^{2} c^{2}}{4}\right ) + x^{3} \left (\frac {A a^{2} d^{2}}{3} + \frac {2 A a b c^{2}}{3} + \frac {2 B a^{2} c d}{3} + \frac {C a^{2} c^{2}}{3}\right ) + x^{2} \left (A a^{2} c d + \frac {B a^{2} c^{2}}{2}\right ) \] Input:
integrate((d*x+c)**2*(b*x**2+a)**2*(D*x**3+C*x**2+B*x+A),x)
Output:
A*a**2*c**2*x + D*b**2*d**2*x**10/10 + x**9*(C*b**2*d**2/9 + 2*D*b**2*c*d/ 9) + x**8*(B*b**2*d**2/8 + C*b**2*c*d/4 + D*a*b*d**2/4 + D*b**2*c**2/8) + x**7*(A*b**2*d**2/7 + 2*B*b**2*c*d/7 + 2*C*a*b*d**2/7 + C*b**2*c**2/7 + 4* D*a*b*c*d/7) + x**6*(A*b**2*c*d/3 + B*a*b*d**2/3 + B*b**2*c**2/6 + 2*C*a*b *c*d/3 + D*a**2*d**2/6 + D*a*b*c**2/3) + x**5*(2*A*a*b*d**2/5 + A*b**2*c** 2/5 + 4*B*a*b*c*d/5 + C*a**2*d**2/5 + 2*C*a*b*c**2/5 + 2*D*a**2*c*d/5) + x **4*(A*a*b*c*d + B*a**2*d**2/4 + B*a*b*c**2/2 + C*a**2*c*d/2 + D*a**2*c**2 /4) + x**3*(A*a**2*d**2/3 + 2*A*a*b*c**2/3 + 2*B*a**2*c*d/3 + C*a**2*c**2/ 3) + x**2*(A*a**2*c*d + B*a**2*c**2/2)
Time = 0.04 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.16 \[ \int (c+d x)^2 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{10} \, D b^{2} d^{2} x^{10} + \frac {1}{9} \, {\left (2 \, D b^{2} c d + C b^{2} d^{2}\right )} x^{9} + \frac {1}{8} \, {\left (D b^{2} c^{2} + 2 \, C b^{2} c d + {\left (2 \, D a b + B b^{2}\right )} d^{2}\right )} x^{8} + \frac {1}{7} \, {\left (C b^{2} c^{2} + 2 \, {\left (2 \, D a b + B b^{2}\right )} c d + {\left (2 \, C a b + A b^{2}\right )} d^{2}\right )} x^{7} + \frac {1}{6} \, {\left ({\left (2 \, D a b + B b^{2}\right )} c^{2} + 2 \, {\left (2 \, C a b + A b^{2}\right )} c d + {\left (D a^{2} + 2 \, B a b\right )} d^{2}\right )} x^{6} + A a^{2} c^{2} x + \frac {1}{5} \, {\left ({\left (2 \, C a b + A b^{2}\right )} c^{2} + 2 \, {\left (D a^{2} + 2 \, B a b\right )} c d + {\left (C a^{2} + 2 \, A a b\right )} d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{2} d^{2} + {\left (D a^{2} + 2 \, B a b\right )} c^{2} + 2 \, {\left (C a^{2} + 2 \, A a b\right )} c d\right )} x^{4} + \frac {1}{3} \, {\left (2 \, B a^{2} c d + A a^{2} d^{2} + {\left (C a^{2} + 2 \, A a b\right )} c^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} c^{2} + 2 \, A a^{2} c d\right )} x^{2} \] Input:
integrate((d*x+c)^2*(b*x^2+a)^2*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")
Output:
1/10*D*b^2*d^2*x^10 + 1/9*(2*D*b^2*c*d + C*b^2*d^2)*x^9 + 1/8*(D*b^2*c^2 + 2*C*b^2*c*d + (2*D*a*b + B*b^2)*d^2)*x^8 + 1/7*(C*b^2*c^2 + 2*(2*D*a*b + B*b^2)*c*d + (2*C*a*b + A*b^2)*d^2)*x^7 + 1/6*((2*D*a*b + B*b^2)*c^2 + 2*( 2*C*a*b + A*b^2)*c*d + (D*a^2 + 2*B*a*b)*d^2)*x^6 + A*a^2*c^2*x + 1/5*((2* C*a*b + A*b^2)*c^2 + 2*(D*a^2 + 2*B*a*b)*c*d + (C*a^2 + 2*A*a*b)*d^2)*x^5 + 1/4*(B*a^2*d^2 + (D*a^2 + 2*B*a*b)*c^2 + 2*(C*a^2 + 2*A*a*b)*c*d)*x^4 + 1/3*(2*B*a^2*c*d + A*a^2*d^2 + (C*a^2 + 2*A*a*b)*c^2)*x^3 + 1/2*(B*a^2*c^2 + 2*A*a^2*c*d)*x^2
Time = 0.35 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.42 \[ \int (c+d x)^2 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{10} \, D b^{2} d^{2} x^{10} + \frac {2}{9} \, D b^{2} c d x^{9} + \frac {1}{9} \, C b^{2} d^{2} x^{9} + \frac {1}{8} \, D b^{2} c^{2} x^{8} + \frac {1}{4} \, C b^{2} c d x^{8} + \frac {1}{4} \, D a b d^{2} x^{8} + \frac {1}{8} \, B b^{2} d^{2} x^{8} + \frac {1}{7} \, C b^{2} c^{2} x^{7} + \frac {4}{7} \, D a b c d x^{7} + \frac {2}{7} \, B b^{2} c d x^{7} + \frac {2}{7} \, C a b d^{2} x^{7} + \frac {1}{7} \, A b^{2} d^{2} x^{7} + \frac {1}{3} \, D a b c^{2} x^{6} + \frac {1}{6} \, B b^{2} c^{2} x^{6} + \frac {2}{3} \, C a b c d x^{6} + \frac {1}{3} \, A b^{2} c d x^{6} + \frac {1}{6} \, D a^{2} d^{2} x^{6} + \frac {1}{3} \, B a b d^{2} x^{6} + \frac {2}{5} \, C a b c^{2} x^{5} + \frac {1}{5} \, A b^{2} c^{2} x^{5} + \frac {2}{5} \, D a^{2} c d x^{5} + \frac {4}{5} \, B a b c d x^{5} + \frac {1}{5} \, C a^{2} d^{2} x^{5} + \frac {2}{5} \, A a b d^{2} x^{5} + \frac {1}{4} \, D a^{2} c^{2} x^{4} + \frac {1}{2} \, B a b c^{2} x^{4} + \frac {1}{2} \, C a^{2} c d x^{4} + A a b c d x^{4} + \frac {1}{4} \, B a^{2} d^{2} x^{4} + \frac {1}{3} \, C a^{2} c^{2} x^{3} + \frac {2}{3} \, A a b c^{2} x^{3} + \frac {2}{3} \, B a^{2} c d x^{3} + \frac {1}{3} \, A a^{2} d^{2} x^{3} + \frac {1}{2} \, B a^{2} c^{2} x^{2} + A a^{2} c d x^{2} + A a^{2} c^{2} x \] Input:
integrate((d*x+c)^2*(b*x^2+a)^2*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")
Output:
1/10*D*b^2*d^2*x^10 + 2/9*D*b^2*c*d*x^9 + 1/9*C*b^2*d^2*x^9 + 1/8*D*b^2*c^ 2*x^8 + 1/4*C*b^2*c*d*x^8 + 1/4*D*a*b*d^2*x^8 + 1/8*B*b^2*d^2*x^8 + 1/7*C* b^2*c^2*x^7 + 4/7*D*a*b*c*d*x^7 + 2/7*B*b^2*c*d*x^7 + 2/7*C*a*b*d^2*x^7 + 1/7*A*b^2*d^2*x^7 + 1/3*D*a*b*c^2*x^6 + 1/6*B*b^2*c^2*x^6 + 2/3*C*a*b*c*d* x^6 + 1/3*A*b^2*c*d*x^6 + 1/6*D*a^2*d^2*x^6 + 1/3*B*a*b*d^2*x^6 + 2/5*C*a* b*c^2*x^5 + 1/5*A*b^2*c^2*x^5 + 2/5*D*a^2*c*d*x^5 + 4/5*B*a*b*c*d*x^5 + 1/ 5*C*a^2*d^2*x^5 + 2/5*A*a*b*d^2*x^5 + 1/4*D*a^2*c^2*x^4 + 1/2*B*a*b*c^2*x^ 4 + 1/2*C*a^2*c*d*x^4 + A*a*b*c*d*x^4 + 1/4*B*a^2*d^2*x^4 + 1/3*C*a^2*c^2* x^3 + 2/3*A*a*b*c^2*x^3 + 2/3*B*a^2*c*d*x^3 + 1/3*A*a^2*d^2*x^3 + 1/2*B*a^ 2*c^2*x^2 + A*a^2*c*d*x^2 + A*a^2*c^2*x
Time = 20.97 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.42 \[ \int (c+d x)^2 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {a^2\,c^2\,x^4\,D}{4}+\frac {a^2\,d^2\,x^6\,D}{6}+\frac {b^2\,c^2\,x^8\,D}{8}+\frac {b^2\,d^2\,x^{10}\,D}{10}+A\,a^2\,c^2\,x+\frac {B\,a^2\,c^2\,x^2}{2}+\frac {A\,a^2\,d^2\,x^3}{3}+\frac {A\,b^2\,c^2\,x^5}{5}+\frac {C\,a^2\,c^2\,x^3}{3}+\frac {B\,a^2\,d^2\,x^4}{4}+\frac {B\,b^2\,c^2\,x^6}{6}+\frac {A\,b^2\,d^2\,x^7}{7}+\frac {C\,a^2\,d^2\,x^5}{5}+\frac {C\,b^2\,c^2\,x^7}{7}+\frac {B\,b^2\,d^2\,x^8}{8}+\frac {C\,b^2\,d^2\,x^9}{9}+\frac {2\,B\,a^2\,c\,d\,x^3}{3}+\frac {A\,b^2\,c\,d\,x^6}{3}+\frac {2\,C\,a\,b\,d^2\,x^7}{7}+\frac {C\,a^2\,c\,d\,x^4}{2}+\frac {2\,B\,b^2\,c\,d\,x^7}{7}+\frac {C\,b^2\,c\,d\,x^8}{4}+\frac {a\,b\,c^2\,x^6\,D}{3}+\frac {a\,b\,d^2\,x^8\,D}{4}+\frac {2\,a^2\,c\,d\,x^5\,D}{5}+\frac {2\,b^2\,c\,d\,x^9\,D}{9}+\frac {2\,A\,a\,b\,c^2\,x^3}{3}+\frac {B\,a\,b\,c^2\,x^4}{2}+\frac {2\,A\,a\,b\,d^2\,x^5}{5}+A\,a^2\,c\,d\,x^2+\frac {2\,C\,a\,b\,c^2\,x^5}{5}+\frac {B\,a\,b\,d^2\,x^6}{3}+A\,a\,b\,c\,d\,x^4+\frac {4\,B\,a\,b\,c\,d\,x^5}{5}+\frac {2\,C\,a\,b\,c\,d\,x^6}{3}+\frac {4\,a\,b\,c\,d\,x^7\,D}{7} \] Input:
int((a + b*x^2)^2*(c + d*x)^2*(A + B*x + C*x^2 + x^3*D),x)
Output:
(a^2*c^2*x^4*D)/4 + (a^2*d^2*x^6*D)/6 + (b^2*c^2*x^8*D)/8 + (b^2*d^2*x^10* D)/10 + A*a^2*c^2*x + (B*a^2*c^2*x^2)/2 + (A*a^2*d^2*x^3)/3 + (A*b^2*c^2*x ^5)/5 + (C*a^2*c^2*x^3)/3 + (B*a^2*d^2*x^4)/4 + (B*b^2*c^2*x^6)/6 + (A*b^2 *d^2*x^7)/7 + (C*a^2*d^2*x^5)/5 + (C*b^2*c^2*x^7)/7 + (B*b^2*d^2*x^8)/8 + (C*b^2*d^2*x^9)/9 + (2*B*a^2*c*d*x^3)/3 + (A*b^2*c*d*x^6)/3 + (2*C*a*b*d^2 *x^7)/7 + (C*a^2*c*d*x^4)/2 + (2*B*b^2*c*d*x^7)/7 + (C*b^2*c*d*x^8)/4 + (a *b*c^2*x^6*D)/3 + (a*b*d^2*x^8*D)/4 + (2*a^2*c*d*x^5*D)/5 + (2*b^2*c*d*x^9 *D)/9 + (2*A*a*b*c^2*x^3)/3 + (B*a*b*c^2*x^4)/2 + (2*A*a*b*d^2*x^5)/5 + A* a^2*c*d*x^2 + (2*C*a*b*c^2*x^5)/5 + (B*a*b*d^2*x^6)/3 + A*a*b*c*d*x^4 + (4 *B*a*b*c*d*x^5)/5 + (2*C*a*b*c*d*x^6)/3 + (4*a*b*c*d*x^7*D)/7
Time = 0.16 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.18 \[ \int (c+d x)^2 \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {x \left (84 b^{2} d^{3} x^{9}+280 b^{2} c \,d^{2} x^{8}+210 a b \,d^{3} x^{7}+105 b^{3} d^{2} x^{7}+315 b^{2} c^{2} d \,x^{7}+120 a \,b^{2} d^{2} x^{6}+720 a b c \,d^{2} x^{6}+240 b^{3} c d \,x^{6}+120 b^{2} c^{3} x^{6}+140 a^{2} d^{3} x^{5}+280 a \,b^{2} c d \,x^{5}+280 a \,b^{2} d^{2} x^{5}+840 a b \,c^{2} d \,x^{5}+140 b^{3} c^{2} x^{5}+336 a^{2} b \,d^{2} x^{4}+504 a^{2} c \,d^{2} x^{4}+168 a \,b^{2} c^{2} x^{4}+672 a \,b^{2} c d \,x^{4}+336 a b \,c^{3} x^{4}+840 a^{2} b c d \,x^{3}+210 a^{2} b \,d^{2} x^{3}+630 a^{2} c^{2} d \,x^{3}+420 a \,b^{2} c^{2} x^{3}+280 a^{3} d^{2} x^{2}+560 a^{2} b \,c^{2} x^{2}+560 a^{2} b c d \,x^{2}+280 a^{2} c^{3} x^{2}+840 a^{3} c d x +420 a^{2} b \,c^{2} x +840 a^{3} c^{2}\right )}{840} \] Input:
int((d*x+c)^2*(b*x^2+a)^2*(D*x^3+C*x^2+B*x+A),x)
Output:
(x*(840*a**3*c**2 + 840*a**3*c*d*x + 280*a**3*d**2*x**2 + 560*a**2*b*c**2* x**2 + 420*a**2*b*c**2*x + 840*a**2*b*c*d*x**3 + 560*a**2*b*c*d*x**2 + 336 *a**2*b*d**2*x**4 + 210*a**2*b*d**2*x**3 + 280*a**2*c**3*x**2 + 630*a**2*c **2*d*x**3 + 504*a**2*c*d**2*x**4 + 140*a**2*d**3*x**5 + 168*a*b**2*c**2*x **4 + 420*a*b**2*c**2*x**3 + 280*a*b**2*c*d*x**5 + 672*a*b**2*c*d*x**4 + 1 20*a*b**2*d**2*x**6 + 280*a*b**2*d**2*x**5 + 336*a*b*c**3*x**4 + 840*a*b*c **2*d*x**5 + 720*a*b*c*d**2*x**6 + 210*a*b*d**3*x**7 + 140*b**3*c**2*x**5 + 240*b**3*c*d*x**6 + 105*b**3*d**2*x**7 + 120*b**2*c**3*x**6 + 315*b**2*c **2*d*x**7 + 280*b**2*c*d**2*x**8 + 84*b**2*d**3*x**9))/840