Integrand size = 20, antiderivative size = 254 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=a^4 A x+2 a^3 A b x^2+\frac {1}{3} a^2 \left (6 A b^2+4 a A c+a^2 C\right ) x^3+a b \left (A \left (b^2+3 a c\right )+a^2 C\right ) x^4+\frac {1}{5} \left (A \left (b^4+12 a b^2 c+6 a^2 c^2\right )+2 a^2 \left (3 b^2+2 a c\right ) C\right ) x^5+\frac {2}{3} b \left (b^2+3 a c\right ) (A c+a C) x^6+\frac {1}{7} \left (2 A c^2 \left (3 b^2+2 a c\right )+\left (b^4+12 a b^2 c+6 a^2 c^2\right ) C\right ) x^7+\frac {1}{2} b c \left (A c^2+\left (b^2+3 a c\right ) C\right ) x^8+\frac {1}{9} c^2 \left (A c^2+6 b^2 C+4 a c C\right ) x^9+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11} \] Output:
a^4*A*x+2*a^3*A*b*x^2+1/3*a^2*(4*A*a*c+6*A*b^2+C*a^2)*x^3+a*b*(A*(3*a*c+b^ 2)+C*a^2)*x^4+1/5*(A*(6*a^2*c^2+12*a*b^2*c+b^4)+2*a^2*(2*a*c+3*b^2)*C)*x^5 +2/3*b*(3*a*c+b^2)*(A*c+C*a)*x^6+1/7*(2*A*c^2*(2*a*c+3*b^2)+(6*a^2*c^2+12* a*b^2*c+b^4)*C)*x^7+1/2*b*c*(A*c^2+(3*a*c+b^2)*C)*x^8+1/9*c^2*(A*c^2+4*C*a *c+6*C*b^2)*x^9+2/5*b*c^3*C*x^10+1/11*c^4*C*x^11
Time = 0.10 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.01 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=a^4 A x+2 a^3 A b x^2+\frac {1}{3} a^2 \left (6 A b^2+4 a A c+a^2 C\right ) x^3+a b \left (A b^2+3 a A c+a^2 C\right ) x^4+\frac {1}{5} \left (A b^4+12 a A b^2 c+6 a^2 A c^2+6 a^2 b^2 C+4 a^3 c C\right ) x^5+\frac {2}{3} b \left (b^2+3 a c\right ) (A c+a C) x^6+\frac {1}{7} \left (6 A b^2 c^2+4 a A c^3+b^4 C+12 a b^2 c C+6 a^2 c^2 C\right ) x^7+\frac {1}{2} b c \left (A c^2+b^2 C+3 a c C\right ) x^8+\frac {1}{9} c^2 \left (A c^2+6 b^2 C+4 a c C\right ) x^9+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11} \] Input:
Integrate[(a + b*x + c*x^2)^4*(A + C*x^2),x]
Output:
a^4*A*x + 2*a^3*A*b*x^2 + (a^2*(6*A*b^2 + 4*a*A*c + a^2*C)*x^3)/3 + a*b*(A *b^2 + 3*a*A*c + a^2*C)*x^4 + ((A*b^4 + 12*a*A*b^2*c + 6*a^2*A*c^2 + 6*a^2 *b^2*C + 4*a^3*c*C)*x^5)/5 + (2*b*(b^2 + 3*a*c)*(A*c + a*C)*x^6)/3 + ((6*A *b^2*c^2 + 4*a*A*c^3 + b^4*C + 12*a*b^2*c*C + 6*a^2*c^2*C)*x^7)/7 + (b*c*( A*c^2 + b^2*C + 3*a*c*C)*x^8)/2 + (c^2*(A*c^2 + 6*b^2*C + 4*a*c*C)*x^9)/9 + (2*b*c^3*C*x^10)/5 + (c^4*C*x^11)/11
Time = 0.58 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2188, 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.
| \(\displaystyle \int \left (A+C x^2\right ) \left (a+b x+c x^2\right )^4 \, dx\) |
| \(\Big \downarrow \) 2188 |
| \(\displaystyle \int \left (a^4 A+4 a^3 A b x+4 a b x^3 \left (a^2 C+A \left (3 a c+b^2\right )\right )+a^2 x^2 \left (a^2 C+4 a A c+6 A b^2\right )+x^6 \left (C \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 A c^2 \left (2 a c+3 b^2\right )\right )+x^4 \left (A \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 a^2 C \left (2 a c+3 b^2\right )\right )+c^2 x^8 \left (4 a c C+A c^2+6 b^2 C\right )+4 b c x^7 \left (C \left (3 a c+b^2\right )+A c^2\right )+4 b x^5 \left (3 a c+b^2\right ) (a C+A c)+4 b c^3 C x^9+c^4 C x^{10}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^4 A x+2 a^3 A b x^2+a b x^4 \left (a^2 C+A \left (3 a c+b^2\right )\right )+\frac {1}{3} a^2 x^3 \left (a^2 C+4 a A c+6 A b^2\right )+\frac {1}{7} x^7 \left (C \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 A c^2 \left (2 a c+3 b^2\right )\right )+\frac {1}{5} x^5 \left (A \left (6 a^2 c^2+12 a b^2 c+b^4\right )+2 a^2 C \left (2 a c+3 b^2\right )\right )+\frac {1}{9} c^2 x^9 \left (4 a c C+A c^2+6 b^2 C\right )+\frac {1}{2} b c x^8 \left (C \left (3 a c+b^2\right )+A c^2\right )+\frac {2}{3} b x^6 \left (3 a c+b^2\right ) (a C+A c)+\frac {2}{5} b c^3 C x^{10}+\frac {1}{11} c^4 C x^{11}\) |
Input:
Int[(a + b*x + c*x^2)^4*(A + C*x^2),x]
Output:
a^4*A*x + 2*a^3*A*b*x^2 + (a^2*(6*A*b^2 + 4*a*A*c + a^2*C)*x^3)/3 + a*b*(A *(b^2 + 3*a*c) + a^2*C)*x^4 + ((A*(b^4 + 12*a*b^2*c + 6*a^2*c^2) + 2*a^2*( 3*b^2 + 2*a*c)*C)*x^5)/5 + (2*b*(b^2 + 3*a*c)*(A*c + a*C)*x^6)/3 + ((2*A*c ^2*(3*b^2 + 2*a*c) + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*C)*x^7)/7 + (b*c*(A*c^ 2 + (b^2 + 3*a*c)*C)*x^8)/2 + (c^2*(A*c^2 + 6*b^2*C + 4*a*c*C)*x^9)/9 + (2 *b*c^3*C*x^10)/5 + (c^4*C*x^11)/11
Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq , x] && IGtQ[p, -2]
Time = 1.41 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.05
| method | result | size |
| norman | \(\frac {c^{4} C \,x^{11}}{11}+\frac {2 b \,c^{3} C \,x^{10}}{5}+\left (\frac {1}{9} c^{4} A +\frac {4}{9} C a \,c^{3}+\frac {2}{3} C \,b^{2} c^{2}\right ) x^{9}+\left (\frac {1}{2} A b \,c^{3}+\frac {3}{2} C a b \,c^{2}+\frac {1}{2} C \,b^{3} c \right ) x^{8}+\left (\frac {4}{7} A a \,c^{3}+\frac {6}{7} A \,b^{2} c^{2}+\frac {6}{7} C \,a^{2} c^{2}+\frac {12}{7} C a \,b^{2} c +\frac {1}{7} C \,b^{4}\right ) x^{7}+\left (2 A a b \,c^{2}+\frac {2}{3} A \,b^{3} c +2 C \,a^{2} b c +\frac {2}{3} C a \,b^{3}\right ) x^{6}+\left (\frac {6}{5} a^{2} A \,c^{2}+\frac {12}{5} A a \,b^{2} c +\frac {1}{5} A \,b^{4}+\frac {4}{5} C \,a^{3} c +\frac {6}{5} C \,a^{2} b^{2}\right ) x^{5}+\left (3 A \,a^{2} b c +A a \,b^{3}+C \,a^{3} b \right ) x^{4}+\left (\frac {4}{3} a^{3} A c +2 a^{2} A \,b^{2}+\frac {1}{3} a^{4} C \right ) x^{3}+2 a^{3} A b \,x^{2}+a^{4} A x\) | \(266\) |
| gosper | \(\frac {1}{2} x^{8} C \,b^{3} c +\frac {4}{5} x^{5} C \,a^{3} c +\frac {6}{5} x^{5} C \,a^{2} b^{2}+\frac {6}{7} x^{7} C \,a^{2} c^{2}+\frac {1}{2} x^{8} A b \,c^{3}+\frac {12}{5} x^{5} A a \,b^{2} c +3 A \,a^{2} b c \,x^{4}+2 x^{6} C \,a^{2} b c +2 x^{6} A a b \,c^{2}+\frac {3}{2} x^{8} C a b \,c^{2}+\frac {12}{7} x^{7} C a \,b^{2} c +2 a^{3} A b \,x^{2}+\frac {2}{3} x^{9} C \,b^{2} c^{2}+\frac {4}{9} x^{9} C a \,c^{3}+\frac {6}{7} x^{7} A \,b^{2} c^{2}+C \,a^{3} b \,x^{4}+\frac {4}{7} x^{7} A a \,c^{3}+\frac {6}{5} x^{5} a^{2} A \,c^{2}+\frac {2}{3} x^{6} C a \,b^{3}+\frac {2}{3} x^{6} A \,b^{3} c +2 A \,a^{2} b^{2} x^{3}+\frac {4}{3} x^{3} a^{3} A c +A a \,b^{3} x^{4}+\frac {1}{5} A \,b^{4} x^{5}+a^{4} A x +\frac {2}{5} b \,c^{3} C \,x^{10}+\frac {1}{3} x^{3} a^{4} C +\frac {1}{9} x^{9} c^{4} A +\frac {1}{7} x^{7} C \,b^{4}+\frac {1}{11} c^{4} C \,x^{11}\) | \(309\) |
| risch | \(\frac {1}{2} x^{8} C \,b^{3} c +\frac {4}{5} x^{5} C \,a^{3} c +\frac {6}{5} x^{5} C \,a^{2} b^{2}+\frac {6}{7} x^{7} C \,a^{2} c^{2}+\frac {1}{2} x^{8} A b \,c^{3}+\frac {12}{5} x^{5} A a \,b^{2} c +3 A \,a^{2} b c \,x^{4}+2 x^{6} C \,a^{2} b c +2 x^{6} A a b \,c^{2}+\frac {3}{2} x^{8} C a b \,c^{2}+\frac {12}{7} x^{7} C a \,b^{2} c +2 a^{3} A b \,x^{2}+\frac {2}{3} x^{9} C \,b^{2} c^{2}+\frac {4}{9} x^{9} C a \,c^{3}+\frac {6}{7} x^{7} A \,b^{2} c^{2}+C \,a^{3} b \,x^{4}+\frac {4}{7} x^{7} A a \,c^{3}+\frac {6}{5} x^{5} a^{2} A \,c^{2}+\frac {2}{3} x^{6} C a \,b^{3}+\frac {2}{3} x^{6} A \,b^{3} c +2 A \,a^{2} b^{2} x^{3}+\frac {4}{3} x^{3} a^{3} A c +A a \,b^{3} x^{4}+\frac {1}{5} A \,b^{4} x^{5}+a^{4} A x +\frac {2}{5} b \,c^{3} C \,x^{10}+\frac {1}{3} x^{3} a^{4} C +\frac {1}{9} x^{9} c^{4} A +\frac {1}{7} x^{7} C \,b^{4}+\frac {1}{11} c^{4} C \,x^{11}\) | \(309\) |
| parallelrisch | \(\frac {1}{2} x^{8} C \,b^{3} c +\frac {4}{5} x^{5} C \,a^{3} c +\frac {6}{5} x^{5} C \,a^{2} b^{2}+\frac {6}{7} x^{7} C \,a^{2} c^{2}+\frac {1}{2} x^{8} A b \,c^{3}+\frac {12}{5} x^{5} A a \,b^{2} c +3 A \,a^{2} b c \,x^{4}+2 x^{6} C \,a^{2} b c +2 x^{6} A a b \,c^{2}+\frac {3}{2} x^{8} C a b \,c^{2}+\frac {12}{7} x^{7} C a \,b^{2} c +2 a^{3} A b \,x^{2}+\frac {2}{3} x^{9} C \,b^{2} c^{2}+\frac {4}{9} x^{9} C a \,c^{3}+\frac {6}{7} x^{7} A \,b^{2} c^{2}+C \,a^{3} b \,x^{4}+\frac {4}{7} x^{7} A a \,c^{3}+\frac {6}{5} x^{5} a^{2} A \,c^{2}+\frac {2}{3} x^{6} C a \,b^{3}+\frac {2}{3} x^{6} A \,b^{3} c +2 A \,a^{2} b^{2} x^{3}+\frac {4}{3} x^{3} a^{3} A c +A a \,b^{3} x^{4}+\frac {1}{5} A \,b^{4} x^{5}+a^{4} A x +\frac {2}{5} b \,c^{3} C \,x^{10}+\frac {1}{3} x^{3} a^{4} C +\frac {1}{9} x^{9} c^{4} A +\frac {1}{7} x^{7} C \,b^{4}+\frac {1}{11} c^{4} C \,x^{11}\) | \(309\) |
| orering | \(\frac {x \left (630 c^{4} C \,x^{10}+2772 b \,c^{3} C \,x^{9}+770 A \,c^{4} x^{8}+3080 C a \,c^{3} x^{8}+4620 C \,b^{2} c^{2} x^{8}+3465 A b \,c^{3} x^{7}+10395 C a b \,c^{2} x^{7}+3465 C \,b^{3} c \,x^{7}+3960 A a \,c^{3} x^{6}+5940 A \,b^{2} c^{2} x^{6}+5940 C \,a^{2} c^{2} x^{6}+11880 C a \,b^{2} c \,x^{6}+990 C \,b^{4} x^{6}+13860 A a b \,c^{2} x^{5}+4620 A \,b^{3} c \,x^{5}+13860 C \,a^{2} b c \,x^{5}+4620 C a \,b^{3} x^{5}+8316 A \,a^{2} c^{2} x^{4}+16632 A a \,b^{2} c \,x^{4}+1386 A \,b^{4} x^{4}+5544 C \,a^{3} c \,x^{4}+8316 C \,a^{2} b^{2} x^{4}+20790 A \,a^{2} b c \,x^{3}+6930 A a \,b^{3} x^{3}+6930 C \,a^{3} b \,x^{3}+9240 A \,a^{3} c \,x^{2}+13860 A \,a^{2} b^{2} x^{2}+2310 C \,a^{4} x^{2}+13860 A \,a^{3} b x +6930 a^{4} A \right )}{6930}\) | \(312\) |
| default | \(\frac {c^{4} C \,x^{11}}{11}+\frac {2 b \,c^{3} C \,x^{10}}{5}+\frac {\left (\left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right ) C +c^{4} A \right ) x^{9}}{9}+\frac {\left (\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) C +4 A b \,c^{3}\right ) x^{8}}{8}+\frac {\left (\left (2 a^{2} c^{2}+8 c a \,b^{2}+\left (2 a c +b^{2}\right )^{2}\right ) C +\left (2 \left (2 a c +b^{2}\right ) c^{2}+4 b^{2} c^{2}\right ) A \right ) x^{7}}{7}+\frac {\left (\left (4 c \,a^{2} b +4 a b \left (2 a c +b^{2}\right )\right ) C +\left (4 a b \,c^{2}+4 \left (2 a c +b^{2}\right ) b c \right ) A \right ) x^{6}}{6}+\frac {\left (\left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right ) C +\left (2 a^{2} c^{2}+8 c a \,b^{2}+\left (2 a c +b^{2}\right )^{2}\right ) A \right ) x^{5}}{5}+\frac {\left (4 C \,a^{3} b +\left (4 c \,a^{2} b +4 a b \left (2 a c +b^{2}\right )\right ) A \right ) x^{4}}{4}+\frac {\left (a^{4} C +\left (2 a^{2} \left (2 a c +b^{2}\right )+4 a^{2} b^{2}\right ) A \right ) x^{3}}{3}+2 a^{3} A b \,x^{2}+a^{4} A x\) | \(343\) |
Input:
int((c*x^2+b*x+a)^4*(C*x^2+A),x,method=_RETURNVERBOSE)
Output:
1/11*c^4*C*x^11+2/5*b*c^3*C*x^10+(1/9*c^4*A+4/9*C*a*c^3+2/3*C*b^2*c^2)*x^9 +(1/2*A*b*c^3+3/2*C*a*b*c^2+1/2*C*b^3*c)*x^8+(4/7*A*a*c^3+6/7*A*b^2*c^2+6/ 7*C*a^2*c^2+12/7*C*a*b^2*c+1/7*C*b^4)*x^7+(2*A*a*b*c^2+2/3*A*b^3*c+2*C*a^2 *b*c+2/3*C*a*b^3)*x^6+(6/5*a^2*A*c^2+12/5*A*a*b^2*c+1/5*A*b^4+4/5*C*a^3*c+ 6/5*C*a^2*b^2)*x^5+(3*A*a^2*b*c+A*a*b^3+C*a^3*b)*x^4+(4/3*a^3*A*c+2*a^2*A* b^2+1/3*a^4*C)*x^3+2*a^3*A*b*x^2+a^4*A*x
Time = 0.06 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.04 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=\frac {1}{11} \, C c^{4} x^{11} + \frac {2}{5} \, C b c^{3} x^{10} + \frac {1}{9} \, {\left (6 \, C b^{2} c^{2} + 4 \, C a c^{3} + A c^{4}\right )} x^{9} + \frac {1}{2} \, {\left (C b^{3} c + 3 \, C a b c^{2} + A b c^{3}\right )} x^{8} + \frac {1}{7} \, {\left (C b^{4} + 12 \, C a b^{2} c + 4 \, A a c^{3} + 6 \, {\left (C a^{2} + A b^{2}\right )} c^{2}\right )} x^{7} + 2 \, A a^{3} b x^{2} + \frac {2}{3} \, {\left (C a b^{3} + 3 \, A a b c^{2} + {\left (3 \, C a^{2} b + A b^{3}\right )} c\right )} x^{6} + A a^{4} x + \frac {1}{5} \, {\left (6 \, C a^{2} b^{2} + A b^{4} + 6 \, A a^{2} c^{2} + 4 \, {\left (C a^{3} + 3 \, A a b^{2}\right )} c\right )} x^{5} + {\left (C a^{3} b + A a b^{3} + 3 \, A a^{2} b c\right )} x^{4} + \frac {1}{3} \, {\left (C a^{4} + 6 \, A a^{2} b^{2} + 4 \, A a^{3} c\right )} x^{3} \] Input:
integrate((c*x^2+b*x+a)^4*(C*x^2+A),x, algorithm="fricas")
Output:
1/11*C*c^4*x^11 + 2/5*C*b*c^3*x^10 + 1/9*(6*C*b^2*c^2 + 4*C*a*c^3 + A*c^4) *x^9 + 1/2*(C*b^3*c + 3*C*a*b*c^2 + A*b*c^3)*x^8 + 1/7*(C*b^4 + 12*C*a*b^2 *c + 4*A*a*c^3 + 6*(C*a^2 + A*b^2)*c^2)*x^7 + 2*A*a^3*b*x^2 + 2/3*(C*a*b^3 + 3*A*a*b*c^2 + (3*C*a^2*b + A*b^3)*c)*x^6 + A*a^4*x + 1/5*(6*C*a^2*b^2 + A*b^4 + 6*A*a^2*c^2 + 4*(C*a^3 + 3*A*a*b^2)*c)*x^5 + (C*a^3*b + A*a*b^3 + 3*A*a^2*b*c)*x^4 + 1/3*(C*a^4 + 6*A*a^2*b^2 + 4*A*a^3*c)*x^3
Time = 0.04 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.26 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=A a^{4} x + 2 A a^{3} b x^{2} + \frac {2 C b c^{3} x^{10}}{5} + \frac {C c^{4} x^{11}}{11} + x^{9} \left (\frac {A c^{4}}{9} + \frac {4 C a c^{3}}{9} + \frac {2 C b^{2} c^{2}}{3}\right ) + x^{8} \left (\frac {A b c^{3}}{2} + \frac {3 C a b c^{2}}{2} + \frac {C b^{3} c}{2}\right ) + x^{7} \cdot \left (\frac {4 A a c^{3}}{7} + \frac {6 A b^{2} c^{2}}{7} + \frac {6 C a^{2} c^{2}}{7} + \frac {12 C a b^{2} c}{7} + \frac {C b^{4}}{7}\right ) + x^{6} \cdot \left (2 A a b c^{2} + \frac {2 A b^{3} c}{3} + 2 C a^{2} b c + \frac {2 C a b^{3}}{3}\right ) + x^{5} \cdot \left (\frac {6 A a^{2} c^{2}}{5} + \frac {12 A a b^{2} c}{5} + \frac {A b^{4}}{5} + \frac {4 C a^{3} c}{5} + \frac {6 C a^{2} b^{2}}{5}\right ) + x^{4} \cdot \left (3 A a^{2} b c + A a b^{3} + C a^{3} b\right ) + x^{3} \cdot \left (\frac {4 A a^{3} c}{3} + 2 A a^{2} b^{2} + \frac {C a^{4}}{3}\right ) \] Input:
integrate((c*x**2+b*x+a)**4*(C*x**2+A),x)
Output:
A*a**4*x + 2*A*a**3*b*x**2 + 2*C*b*c**3*x**10/5 + C*c**4*x**11/11 + x**9*( A*c**4/9 + 4*C*a*c**3/9 + 2*C*b**2*c**2/3) + x**8*(A*b*c**3/2 + 3*C*a*b*c* *2/2 + C*b**3*c/2) + x**7*(4*A*a*c**3/7 + 6*A*b**2*c**2/7 + 6*C*a**2*c**2/ 7 + 12*C*a*b**2*c/7 + C*b**4/7) + x**6*(2*A*a*b*c**2 + 2*A*b**3*c/3 + 2*C* a**2*b*c + 2*C*a*b**3/3) + x**5*(6*A*a**2*c**2/5 + 12*A*a*b**2*c/5 + A*b** 4/5 + 4*C*a**3*c/5 + 6*C*a**2*b**2/5) + x**4*(3*A*a**2*b*c + A*a*b**3 + C* a**3*b) + x**3*(4*A*a**3*c/3 + 2*A*a**2*b**2 + C*a**4/3)
Time = 0.03 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.04 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=\frac {1}{11} \, C c^{4} x^{11} + \frac {2}{5} \, C b c^{3} x^{10} + \frac {1}{9} \, {\left (6 \, C b^{2} c^{2} + 4 \, C a c^{3} + A c^{4}\right )} x^{9} + \frac {1}{2} \, {\left (C b^{3} c + 3 \, C a b c^{2} + A b c^{3}\right )} x^{8} + \frac {1}{7} \, {\left (C b^{4} + 12 \, C a b^{2} c + 4 \, A a c^{3} + 6 \, {\left (C a^{2} + A b^{2}\right )} c^{2}\right )} x^{7} + 2 \, A a^{3} b x^{2} + \frac {2}{3} \, {\left (C a b^{3} + 3 \, A a b c^{2} + {\left (3 \, C a^{2} b + A b^{3}\right )} c\right )} x^{6} + A a^{4} x + \frac {1}{5} \, {\left (6 \, C a^{2} b^{2} + A b^{4} + 6 \, A a^{2} c^{2} + 4 \, {\left (C a^{3} + 3 \, A a b^{2}\right )} c\right )} x^{5} + {\left (C a^{3} b + A a b^{3} + 3 \, A a^{2} b c\right )} x^{4} + \frac {1}{3} \, {\left (C a^{4} + 6 \, A a^{2} b^{2} + 4 \, A a^{3} c\right )} x^{3} \] Input:
integrate((c*x^2+b*x+a)^4*(C*x^2+A),x, algorithm="maxima")
Output:
1/11*C*c^4*x^11 + 2/5*C*b*c^3*x^10 + 1/9*(6*C*b^2*c^2 + 4*C*a*c^3 + A*c^4) *x^9 + 1/2*(C*b^3*c + 3*C*a*b*c^2 + A*b*c^3)*x^8 + 1/7*(C*b^4 + 12*C*a*b^2 *c + 4*A*a*c^3 + 6*(C*a^2 + A*b^2)*c^2)*x^7 + 2*A*a^3*b*x^2 + 2/3*(C*a*b^3 + 3*A*a*b*c^2 + (3*C*a^2*b + A*b^3)*c)*x^6 + A*a^4*x + 1/5*(6*C*a^2*b^2 + A*b^4 + 6*A*a^2*c^2 + 4*(C*a^3 + 3*A*a*b^2)*c)*x^5 + (C*a^3*b + A*a*b^3 + 3*A*a^2*b*c)*x^4 + 1/3*(C*a^4 + 6*A*a^2*b^2 + 4*A*a^3*c)*x^3
Time = 0.32 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.21 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=\frac {1}{11} \, C c^{4} x^{11} + \frac {2}{5} \, C b c^{3} x^{10} + \frac {2}{3} \, C b^{2} c^{2} x^{9} + \frac {4}{9} \, C a c^{3} x^{9} + \frac {1}{9} \, A c^{4} x^{9} + \frac {1}{2} \, C b^{3} c x^{8} + \frac {3}{2} \, C a b c^{2} x^{8} + \frac {1}{2} \, A b c^{3} x^{8} + \frac {1}{7} \, C b^{4} x^{7} + \frac {12}{7} \, C a b^{2} c x^{7} + \frac {6}{7} \, C a^{2} c^{2} x^{7} + \frac {6}{7} \, A b^{2} c^{2} x^{7} + \frac {4}{7} \, A a c^{3} x^{7} + \frac {2}{3} \, C a b^{3} x^{6} + 2 \, C a^{2} b c x^{6} + \frac {2}{3} \, A b^{3} c x^{6} + 2 \, A a b c^{2} x^{6} + \frac {6}{5} \, C a^{2} b^{2} x^{5} + \frac {1}{5} \, A b^{4} x^{5} + \frac {4}{5} \, C a^{3} c x^{5} + \frac {12}{5} \, A a b^{2} c x^{5} + \frac {6}{5} \, A a^{2} c^{2} x^{5} + C a^{3} b x^{4} + A a b^{3} x^{4} + 3 \, A a^{2} b c x^{4} + \frac {1}{3} \, C a^{4} x^{3} + 2 \, A a^{2} b^{2} x^{3} + \frac {4}{3} \, A a^{3} c x^{3} + 2 \, A a^{3} b x^{2} + A a^{4} x \] Input:
integrate((c*x^2+b*x+a)^4*(C*x^2+A),x, algorithm="giac")
Output:
1/11*C*c^4*x^11 + 2/5*C*b*c^3*x^10 + 2/3*C*b^2*c^2*x^9 + 4/9*C*a*c^3*x^9 + 1/9*A*c^4*x^9 + 1/2*C*b^3*c*x^8 + 3/2*C*a*b*c^2*x^8 + 1/2*A*b*c^3*x^8 + 1 /7*C*b^4*x^7 + 12/7*C*a*b^2*c*x^7 + 6/7*C*a^2*c^2*x^7 + 6/7*A*b^2*c^2*x^7 + 4/7*A*a*c^3*x^7 + 2/3*C*a*b^3*x^6 + 2*C*a^2*b*c*x^6 + 2/3*A*b^3*c*x^6 + 2*A*a*b*c^2*x^6 + 6/5*C*a^2*b^2*x^5 + 1/5*A*b^4*x^5 + 4/5*C*a^3*c*x^5 + 12 /5*A*a*b^2*c*x^5 + 6/5*A*a^2*c^2*x^5 + C*a^3*b*x^4 + A*a*b^3*x^4 + 3*A*a^2 *b*c*x^4 + 1/3*C*a^4*x^3 + 2*A*a^2*b^2*x^3 + 4/3*A*a^3*c*x^3 + 2*A*a^3*b*x ^2 + A*a^4*x
Time = 0.16 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.96 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=x^5\,\left (\frac {4\,C\,a^3\,c}{5}+\frac {6\,C\,a^2\,b^2}{5}+\frac {6\,A\,a^2\,c^2}{5}+\frac {12\,A\,a\,b^2\,c}{5}+\frac {A\,b^4}{5}\right )+x^7\,\left (\frac {6\,C\,a^2\,c^2}{7}+\frac {12\,C\,a\,b^2\,c}{7}+\frac {4\,A\,a\,c^3}{7}+\frac {C\,b^4}{7}+\frac {6\,A\,b^2\,c^2}{7}\right )+x^3\,\left (\frac {C\,a^4}{3}+\frac {4\,A\,c\,a^3}{3}+2\,A\,a^2\,b^2\right )+x^9\,\left (\frac {2\,C\,b^2\,c^2}{3}+\frac {A\,c^4}{9}+\frac {4\,C\,a\,c^3}{9}\right )+\frac {C\,c^4\,x^{11}}{11}+A\,a^4\,x+\frac {2\,b\,x^6\,\left (b^2+3\,a\,c\right )\,\left (A\,c+C\,a\right )}{3}+a\,b\,x^4\,\left (C\,a^2+3\,A\,c\,a+A\,b^2\right )+\frac {b\,c\,x^8\,\left (C\,b^2+A\,c^2+3\,C\,a\,c\right )}{2}+2\,A\,a^3\,b\,x^2+\frac {2\,C\,b\,c^3\,x^{10}}{5} \] Input:
int((A + C*x^2)*(a + b*x + c*x^2)^4,x)
Output:
x^5*((A*b^4)/5 + (6*A*a^2*c^2)/5 + (6*C*a^2*b^2)/5 + (4*C*a^3*c)/5 + (12*A *a*b^2*c)/5) + x^7*((C*b^4)/7 + (6*A*b^2*c^2)/7 + (6*C*a^2*c^2)/7 + (4*A*a *c^3)/7 + (12*C*a*b^2*c)/7) + x^3*((C*a^4)/3 + 2*A*a^2*b^2 + (4*A*a^3*c)/3 ) + x^9*((A*c^4)/9 + (2*C*b^2*c^2)/3 + (4*C*a*c^3)/9) + (C*c^4*x^11)/11 + A*a^4*x + (2*b*x^6*(3*a*c + b^2)*(A*c + C*a))/3 + a*b*x^4*(A*b^2 + C*a^2 + 3*A*a*c) + (b*c*x^8*(A*c^2 + C*b^2 + 3*C*a*c))/2 + 2*A*a^3*b*x^2 + (2*C*b *c^3*x^10)/5
Time = 0.16 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.79 \[ \int \left (a+b x+c x^2\right )^4 \left (A+C x^2\right ) \, dx=\frac {x \left (630 c^{5} x^{10}+2772 b \,c^{4} x^{9}+3850 a \,c^{4} x^{8}+4620 b^{2} c^{3} x^{8}+13860 a b \,c^{3} x^{7}+3465 b^{3} c^{2} x^{7}+9900 a^{2} c^{3} x^{6}+17820 a \,b^{2} c^{2} x^{6}+990 b^{4} c \,x^{6}+27720 a^{2} b \,c^{2} x^{5}+9240 a \,b^{3} c \,x^{5}+13860 a^{3} c^{2} x^{4}+24948 a^{2} b^{2} c \,x^{4}+1386 a \,b^{4} x^{4}+27720 a^{3} b c \,x^{3}+6930 a^{2} b^{3} x^{3}+11550 a^{4} c \,x^{2}+13860 a^{3} b^{2} x^{2}+13860 a^{4} b x +6930 a^{5}\right )}{6930} \] Input:
int((c*x^2+b*x+a)^4*(C*x^2+A),x)
Output:
(x*(6930*a**5 + 13860*a**4*b*x + 11550*a**4*c*x**2 + 13860*a**3*b**2*x**2 + 27720*a**3*b*c*x**3 + 13860*a**3*c**2*x**4 + 6930*a**2*b**3*x**3 + 24948 *a**2*b**2*c*x**4 + 27720*a**2*b*c**2*x**5 + 9900*a**2*c**3*x**6 + 1386*a* b**4*x**4 + 9240*a*b**3*c*x**5 + 17820*a*b**2*c**2*x**6 + 13860*a*b*c**3*x **7 + 3850*a*c**4*x**8 + 990*b**4*c*x**6 + 3465*b**3*c**2*x**7 + 4620*b**2 *c**3*x**8 + 2772*b*c**4*x**9 + 630*c**5*x**10))/6930