Integrand size = 22, antiderivative size = 334 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^3 \, dx=-\frac {(B d-A e) \left (c d^2+a e^2\right )^3 (d+e x)^5}{5 e^8}+\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^6}{6 e^8}-\frac {3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^7}{7 e^8}-\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^8}{8 e^8}-\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^9}{9 e^8}+\frac {3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{10}}{10 e^8}-\frac {c^3 (7 B d-A e) (d+e x)^{11}}{11 e^8}+\frac {B c^3 (d+e x)^{12}}{12 e^8} \]
-1/5*(-A*e+B*d)*(a*e^2+c*d^2)^3*(e*x+d)^5/e^8+1/6*(a*e^2+c*d^2)^2*(-6*A*c* d*e+B*a*e^2+7*B*c*d^2)*(e*x+d)^6/e^8-3/7*c*(a*e^2+c*d^2)*(-A*a*e^3-5*A*c*d ^2*e+3*B*a*d*e^2+7*B*c*d^3)*(e*x+d)^7/e^8-1/8*c*(4*A*c*d*e*(3*a*e^2+5*c*d^ 2)-B*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2*d^4))*(e*x+d)^8/e^8-1/9*c^2*(-3*A*a* e^3-15*A*c*d^2*e+15*B*a*d*e^2+35*B*c*d^3)*(e*x+d)^9/e^8+3/10*c^2*(-2*A*c*d *e+B*a*e^2+7*B*c*d^2)*(e*x+d)^10/e^8-1/11*c^3*(-A*e+7*B*d)*(e*x+d)^11/e^8+ 1/12*B*c^3*(e*x+d)^12/e^8
Time = 0.08 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.31 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^3 \, dx=a^3 A d^4 x+\frac {1}{2} a^3 d^3 (B d+4 A e) x^2+\frac {1}{3} a^2 d^2 \left (3 A c d^2+4 a B d e+6 a A e^2\right ) x^3+\frac {1}{4} a^2 d \left (3 B c d^3+12 A c d^2 e+6 a B d e^2+4 a A e^3\right ) x^4+\frac {1}{5} a \left (4 a B d e \left (3 c d^2+a e^2\right )+A \left (3 c^2 d^4+18 a c d^2 e^2+a^2 e^4\right )\right ) x^5+\frac {1}{6} a \left (12 A c d e \left (c d^2+a e^2\right )+B \left (3 c^2 d^4+18 a c d^2 e^2+a^2 e^4\right )\right ) x^6+\frac {1}{7} c \left (12 a B d e \left (c d^2+a e^2\right )+A \left (c^2 d^4+18 a c d^2 e^2+3 a^2 e^4\right )\right ) x^7+\frac {1}{8} c \left (4 A c d e \left (c d^2+3 a e^2\right )+B \left (c^2 d^4+18 a c d^2 e^2+3 a^2 e^4\right )\right ) x^8+\frac {1}{9} c^2 e \left (4 B c d^3+6 A c d^2 e+12 a B d e^2+3 a A e^3\right ) x^9+\frac {1}{10} c^2 e^2 \left (6 B c d^2+4 A c d e+3 a B e^2\right ) x^{10}+\frac {1}{11} c^3 e^3 (4 B d+A e) x^{11}+\frac {1}{12} B c^3 e^4 x^{12} \]
a^3*A*d^4*x + (a^3*d^3*(B*d + 4*A*e)*x^2)/2 + (a^2*d^2*(3*A*c*d^2 + 4*a*B* d*e + 6*a*A*e^2)*x^3)/3 + (a^2*d*(3*B*c*d^3 + 12*A*c*d^2*e + 6*a*B*d*e^2 + 4*a*A*e^3)*x^4)/4 + (a*(4*a*B*d*e*(3*c*d^2 + a*e^2) + A*(3*c^2*d^4 + 18*a *c*d^2*e^2 + a^2*e^4))*x^5)/5 + (a*(12*A*c*d*e*(c*d^2 + a*e^2) + B*(3*c^2* d^4 + 18*a*c*d^2*e^2 + a^2*e^4))*x^6)/6 + (c*(12*a*B*d*e*(c*d^2 + a*e^2) + A*(c^2*d^4 + 18*a*c*d^2*e^2 + 3*a^2*e^4))*x^7)/7 + (c*(4*A*c*d*e*(c*d^2 + 3*a*e^2) + B*(c^2*d^4 + 18*a*c*d^2*e^2 + 3*a^2*e^4))*x^8)/8 + (c^2*e*(4*B *c*d^3 + 6*A*c*d^2*e + 12*a*B*d*e^2 + 3*a*A*e^3)*x^9)/9 + (c^2*e^2*(6*B*c* d^2 + 4*A*c*d*e + 3*a*B*e^2)*x^10)/10 + (c^3*e^3*(4*B*d + A*e)*x^11)/11 + (B*c^3*e^4*x^12)/12
Time = 0.73 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {652, 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.
| \(\displaystyle \int \left (a+c x^2\right )^3 (A+B x) (d+e x)^4 \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {c (d+e x)^7 \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7}-\frac {3 c^2 (d+e x)^9 \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7}+\frac {c^2 (d+e x)^8 \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7}+\frac {(d+e x)^5 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7}+\frac {(d+e x)^4 \left (a e^2+c d^2\right )^3 (A e-B d)}{e^7}+\frac {3 c (d+e x)^6 \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7}+\frac {c^3 (d+e x)^{10} (A e-7 B d)}{e^7}+\frac {B c^3 (d+e x)^{11}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c (d+e x)^8 \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{8 e^8}+\frac {3 c^2 (d+e x)^{10} \left (a B e^2-2 A c d e+7 B c d^2\right )}{10 e^8}-\frac {c^2 (d+e x)^9 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{9 e^8}+\frac {(d+e x)^6 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{6 e^8}-\frac {(d+e x)^5 \left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8}-\frac {3 c (d+e x)^7 \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{7 e^8}-\frac {c^3 (d+e x)^{11} (7 B d-A e)}{11 e^8}+\frac {B c^3 (d+e x)^{12}}{12 e^8}\) |
-1/5*((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^5)/e^8 + ((c*d^2 + a*e^2)^2* (7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*(d + e*x)^6)/(6*e^8) - (3*c*(c*d^2 + a*e ^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^7)/(7*e^8) - (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3* a^2*e^4))*(d + e*x)^8)/(8*e^8) - (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B* d*e^2 - 3*a*A*e^3)*(d + e*x)^9)/(9*e^8) + (3*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^10)/(10*e^8) - (c^3*(7*B*d - A*e)*(d + e*x)^11)/(11*e^8 ) + (B*c^3*(d + e*x)^12)/(12*e^8)
3.14.14.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Time = 0.25 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.36
| method | result | size |
| default | \(\frac {B \,c^{3} e^{4} x^{12}}{12}+\frac {\left (e^{4} A +4 e^{3} d B \right ) c^{3} x^{11}}{11}+\frac {\left (\left (4 d A \,e^{3}+6 B \,d^{2} e^{2}\right ) c^{3}+3 B \,e^{4} a \,c^{2}\right ) x^{10}}{10}+\frac {\left (\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c^{3}+3 \left (e^{4} A +4 e^{3} d B \right ) a \,c^{2}\right ) x^{9}}{9}+\frac {\left (\left (4 A \,d^{3} e +B \,d^{4}\right ) c^{3}+3 \left (4 d A \,e^{3}+6 B \,d^{2} e^{2}\right ) a \,c^{2}+3 B \,e^{4} c \,a^{2}\right ) x^{8}}{8}+\frac {\left (A \,d^{4} c^{3}+3 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a \,c^{2}+3 \left (e^{4} A +4 e^{3} d B \right ) c \,a^{2}\right ) x^{7}}{7}+\frac {\left (3 \left (4 A \,d^{3} e +B \,d^{4}\right ) a \,c^{2}+3 \left (4 d A \,e^{3}+6 B \,d^{2} e^{2}\right ) c \,a^{2}+B \,e^{4} a^{3}\right ) x^{6}}{6}+\frac {\left (3 d^{4} A a \,c^{2}+3 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c \,a^{2}+\left (e^{4} A +4 e^{3} d B \right ) a^{3}\right ) x^{5}}{5}+\frac {\left (3 \left (4 A \,d^{3} e +B \,d^{4}\right ) c \,a^{2}+\left (4 d A \,e^{3}+6 B \,d^{2} e^{2}\right ) a^{3}\right ) x^{4}}{4}+\frac {\left (3 d^{4} A c \,a^{2}+\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{3}\right ) x^{3}}{3}+\frac {\left (4 A \,d^{3} e +B \,d^{4}\right ) a^{3} x^{2}}{2}+d^{4} A \,a^{3} x\) | \(455\) |
| norman | \(\frac {B \,c^{3} e^{4} x^{12}}{12}+\left (\frac {1}{11} A \,c^{3} e^{4}+\frac {4}{11} B \,c^{3} d \,e^{3}\right ) x^{11}+\left (\frac {2}{5} A \,c^{3} d \,e^{3}+\frac {3}{10} B \,e^{4} a \,c^{2}+\frac {3}{5} B \,c^{3} d^{2} e^{2}\right ) x^{10}+\left (\frac {1}{3} A a \,c^{2} e^{4}+\frac {2}{3} A \,c^{3} d^{2} e^{2}+\frac {4}{3} B a \,c^{2} d \,e^{3}+\frac {4}{9} B \,c^{3} d^{3} e \right ) x^{9}+\left (\frac {3}{2} A a \,c^{2} d \,e^{3}+\frac {1}{2} A \,c^{3} d^{3} e +\frac {3}{8} B \,e^{4} c \,a^{2}+\frac {9}{4} B a \,c^{2} d^{2} e^{2}+\frac {1}{8} B \,c^{3} d^{4}\right ) x^{8}+\left (\frac {3}{7} A \,a^{2} c \,e^{4}+\frac {18}{7} A a \,c^{2} d^{2} e^{2}+\frac {1}{7} A \,d^{4} c^{3}+\frac {12}{7} B \,a^{2} c d \,e^{3}+\frac {12}{7} B a \,c^{2} d^{3} e \right ) x^{7}+\left (2 A \,a^{2} c d \,e^{3}+2 A a \,c^{2} d^{3} e +\frac {1}{6} B \,e^{4} a^{3}+3 B \,a^{2} c \,d^{2} e^{2}+\frac {1}{2} B a \,c^{2} d^{4}\right ) x^{6}+\left (\frac {1}{5} A \,a^{3} e^{4}+\frac {18}{5} A \,a^{2} c \,d^{2} e^{2}+\frac {3}{5} d^{4} A a \,c^{2}+\frac {4}{5} B \,a^{3} d \,e^{3}+\frac {12}{5} B \,a^{2} c \,d^{3} e \right ) x^{5}+\left (A \,a^{3} d \,e^{3}+3 A \,a^{2} c \,d^{3} e +\frac {3}{2} B \,a^{3} d^{2} e^{2}+\frac {3}{4} B \,a^{2} c \,d^{4}\right ) x^{4}+\left (2 A \,a^{3} d^{2} e^{2}+d^{4} A c \,a^{2}+\frac {4}{3} B \,a^{3} d^{3} e \right ) x^{3}+\left (2 A \,a^{3} d^{3} e +\frac {1}{2} B \,a^{3} d^{4}\right ) x^{2}+d^{4} A \,a^{3} x\) | \(473\) |
| gosper | \(\frac {1}{12} B \,c^{3} e^{4} x^{12}+3 x^{4} A \,a^{2} c \,d^{3} e +\frac {12}{5} x^{5} B \,a^{2} c \,d^{3} e +\frac {1}{11} x^{11} A \,c^{3} e^{4}+\frac {1}{8} x^{8} B \,c^{3} d^{4}+\frac {1}{7} x^{7} A \,d^{4} c^{3}+\frac {1}{5} x^{5} A \,a^{3} e^{4}+\frac {1}{2} x^{2} B \,a^{3} d^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{3}+d^{4} A \,a^{3} x +\frac {18}{5} x^{5} A \,a^{2} c \,d^{2} e^{2}+3 x^{6} B \,a^{2} c \,d^{2} e^{2}+2 x^{6} A a \,c^{2} d^{3} e +2 x^{6} A \,a^{2} c d \,e^{3}+\frac {12}{7} x^{7} B a \,c^{2} d^{3} e +\frac {4}{3} x^{9} B a \,c^{2} d \,e^{3}+\frac {3}{2} x^{8} A a \,c^{2} d \,e^{3}+\frac {9}{4} x^{8} B a \,c^{2} d^{2} e^{2}+\frac {18}{7} x^{7} A a \,c^{2} d^{2} e^{2}+\frac {12}{7} x^{7} B \,a^{2} c d \,e^{3}+\frac {3}{7} x^{7} A \,a^{2} c \,e^{4}+\frac {3}{2} x^{4} B \,a^{3} d^{2} e^{2}+\frac {3}{5} x^{5} d^{4} A a \,c^{2}+\frac {4}{5} x^{5} B \,a^{3} d \,e^{3}+x^{4} A \,a^{3} d \,e^{3}+\frac {3}{4} x^{4} B \,a^{2} c \,d^{4}+2 x^{3} A \,a^{3} d^{2} e^{2}+x^{3} d^{4} A c \,a^{2}+\frac {4}{3} x^{3} B \,a^{3} d^{3} e +2 x^{2} A \,a^{3} d^{3} e +\frac {3}{8} x^{8} B \,e^{4} c \,a^{2}+\frac {1}{2} x^{6} B a \,c^{2} d^{4}+\frac {3}{10} x^{10} B \,e^{4} a \,c^{2}+\frac {1}{3} x^{9} A a \,c^{2} e^{4}+\frac {2}{3} x^{9} A \,c^{3} d^{2} e^{2}+\frac {4}{9} x^{9} B \,c^{3} d^{3} e +\frac {1}{2} x^{8} A \,c^{3} d^{3} e +\frac {4}{11} x^{11} B \,c^{3} d \,e^{3}+\frac {2}{5} x^{10} A \,c^{3} d \,e^{3}+\frac {3}{5} x^{10} B \,c^{3} d^{2} e^{2}\) | \(537\) |
| risch | \(\frac {1}{12} B \,c^{3} e^{4} x^{12}+3 x^{4} A \,a^{2} c \,d^{3} e +\frac {12}{5} x^{5} B \,a^{2} c \,d^{3} e +\frac {1}{11} x^{11} A \,c^{3} e^{4}+\frac {1}{8} x^{8} B \,c^{3} d^{4}+\frac {1}{7} x^{7} A \,d^{4} c^{3}+\frac {1}{5} x^{5} A \,a^{3} e^{4}+\frac {1}{2} x^{2} B \,a^{3} d^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{3}+d^{4} A \,a^{3} x +\frac {18}{5} x^{5} A \,a^{2} c \,d^{2} e^{2}+3 x^{6} B \,a^{2} c \,d^{2} e^{2}+2 x^{6} A a \,c^{2} d^{3} e +2 x^{6} A \,a^{2} c d \,e^{3}+\frac {12}{7} x^{7} B a \,c^{2} d^{3} e +\frac {4}{3} x^{9} B a \,c^{2} d \,e^{3}+\frac {3}{2} x^{8} A a \,c^{2} d \,e^{3}+\frac {9}{4} x^{8} B a \,c^{2} d^{2} e^{2}+\frac {18}{7} x^{7} A a \,c^{2} d^{2} e^{2}+\frac {12}{7} x^{7} B \,a^{2} c d \,e^{3}+\frac {3}{7} x^{7} A \,a^{2} c \,e^{4}+\frac {3}{2} x^{4} B \,a^{3} d^{2} e^{2}+\frac {3}{5} x^{5} d^{4} A a \,c^{2}+\frac {4}{5} x^{5} B \,a^{3} d \,e^{3}+x^{4} A \,a^{3} d \,e^{3}+\frac {3}{4} x^{4} B \,a^{2} c \,d^{4}+2 x^{3} A \,a^{3} d^{2} e^{2}+x^{3} d^{4} A c \,a^{2}+\frac {4}{3} x^{3} B \,a^{3} d^{3} e +2 x^{2} A \,a^{3} d^{3} e +\frac {3}{8} x^{8} B \,e^{4} c \,a^{2}+\frac {1}{2} x^{6} B a \,c^{2} d^{4}+\frac {3}{10} x^{10} B \,e^{4} a \,c^{2}+\frac {1}{3} x^{9} A a \,c^{2} e^{4}+\frac {2}{3} x^{9} A \,c^{3} d^{2} e^{2}+\frac {4}{9} x^{9} B \,c^{3} d^{3} e +\frac {1}{2} x^{8} A \,c^{3} d^{3} e +\frac {4}{11} x^{11} B \,c^{3} d \,e^{3}+\frac {2}{5} x^{10} A \,c^{3} d \,e^{3}+\frac {3}{5} x^{10} B \,c^{3} d^{2} e^{2}\) | \(537\) |
| parallelrisch | \(\frac {1}{12} B \,c^{3} e^{4} x^{12}+3 x^{4} A \,a^{2} c \,d^{3} e +\frac {12}{5} x^{5} B \,a^{2} c \,d^{3} e +\frac {1}{11} x^{11} A \,c^{3} e^{4}+\frac {1}{8} x^{8} B \,c^{3} d^{4}+\frac {1}{7} x^{7} A \,d^{4} c^{3}+\frac {1}{5} x^{5} A \,a^{3} e^{4}+\frac {1}{2} x^{2} B \,a^{3} d^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{3}+d^{4} A \,a^{3} x +\frac {18}{5} x^{5} A \,a^{2} c \,d^{2} e^{2}+3 x^{6} B \,a^{2} c \,d^{2} e^{2}+2 x^{6} A a \,c^{2} d^{3} e +2 x^{6} A \,a^{2} c d \,e^{3}+\frac {12}{7} x^{7} B a \,c^{2} d^{3} e +\frac {4}{3} x^{9} B a \,c^{2} d \,e^{3}+\frac {3}{2} x^{8} A a \,c^{2} d \,e^{3}+\frac {9}{4} x^{8} B a \,c^{2} d^{2} e^{2}+\frac {18}{7} x^{7} A a \,c^{2} d^{2} e^{2}+\frac {12}{7} x^{7} B \,a^{2} c d \,e^{3}+\frac {3}{7} x^{7} A \,a^{2} c \,e^{4}+\frac {3}{2} x^{4} B \,a^{3} d^{2} e^{2}+\frac {3}{5} x^{5} d^{4} A a \,c^{2}+\frac {4}{5} x^{5} B \,a^{3} d \,e^{3}+x^{4} A \,a^{3} d \,e^{3}+\frac {3}{4} x^{4} B \,a^{2} c \,d^{4}+2 x^{3} A \,a^{3} d^{2} e^{2}+x^{3} d^{4} A c \,a^{2}+\frac {4}{3} x^{3} B \,a^{3} d^{3} e +2 x^{2} A \,a^{3} d^{3} e +\frac {3}{8} x^{8} B \,e^{4} c \,a^{2}+\frac {1}{2} x^{6} B a \,c^{2} d^{4}+\frac {3}{10} x^{10} B \,e^{4} a \,c^{2}+\frac {1}{3} x^{9} A a \,c^{2} e^{4}+\frac {2}{3} x^{9} A \,c^{3} d^{2} e^{2}+\frac {4}{9} x^{9} B \,c^{3} d^{3} e +\frac {1}{2} x^{8} A \,c^{3} d^{3} e +\frac {4}{11} x^{11} B \,c^{3} d \,e^{3}+\frac {2}{5} x^{10} A \,c^{3} d \,e^{3}+\frac {3}{5} x^{10} B \,c^{3} d^{2} e^{2}\) | \(537\) |
1/12*B*c^3*e^4*x^12+1/11*(A*e^4+4*B*d*e^3)*c^3*x^11+1/10*((4*A*d*e^3+6*B*d ^2*e^2)*c^3+3*B*e^4*a*c^2)*x^10+1/9*((6*A*d^2*e^2+4*B*d^3*e)*c^3+3*(A*e^4+ 4*B*d*e^3)*a*c^2)*x^9+1/8*((4*A*d^3*e+B*d^4)*c^3+3*(4*A*d*e^3+6*B*d^2*e^2) *a*c^2+3*B*e^4*c*a^2)*x^8+1/7*(A*d^4*c^3+3*(6*A*d^2*e^2+4*B*d^3*e)*a*c^2+3 *(A*e^4+4*B*d*e^3)*c*a^2)*x^7+1/6*(3*(4*A*d^3*e+B*d^4)*a*c^2+3*(4*A*d*e^3+ 6*B*d^2*e^2)*c*a^2+B*e^4*a^3)*x^6+1/5*(3*d^4*A*a*c^2+3*(6*A*d^2*e^2+4*B*d^ 3*e)*c*a^2+(A*e^4+4*B*d*e^3)*a^3)*x^5+1/4*(3*(4*A*d^3*e+B*d^4)*c*a^2+(4*A* d*e^3+6*B*d^2*e^2)*a^3)*x^4+1/3*(3*d^4*A*c*a^2+(6*A*d^2*e^2+4*B*d^3*e)*a^3 )*x^3+1/2*(4*A*d^3*e+B*d^4)*a^3*x^2+d^4*A*a^3*x
Time = 0.26 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.43 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^3 \, dx=\frac {1}{12} \, B c^{3} e^{4} x^{12} + \frac {1}{11} \, {\left (4 \, B c^{3} d e^{3} + A c^{3} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (6 \, B c^{3} d^{2} e^{2} + 4 \, A c^{3} d e^{3} + 3 \, B a c^{2} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{3} d^{3} e + 6 \, A c^{3} d^{2} e^{2} + 12 \, B a c^{2} d e^{3} + 3 \, A a c^{2} e^{4}\right )} x^{9} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (B c^{3} d^{4} + 4 \, A c^{3} d^{3} e + 18 \, B a c^{2} d^{2} e^{2} + 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (A c^{3} d^{4} + 12 \, B a c^{2} d^{3} e + 18 \, A a c^{2} d^{2} e^{2} + 12 \, B a^{2} c d e^{3} + 3 \, A a^{2} c e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, B a c^{2} d^{4} + 12 \, A a c^{2} d^{3} e + 18 \, B a^{2} c d^{2} e^{2} + 12 \, A a^{2} c d e^{3} + B a^{3} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, A a c^{2} d^{4} + 12 \, B a^{2} c d^{3} e + 18 \, A a^{2} c d^{2} e^{2} + 4 \, B a^{3} d e^{3} + A a^{3} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{4} + 12 \, A a^{2} c d^{3} e + 6 \, B a^{3} d^{2} e^{2} + 4 \, A a^{3} d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} c d^{4} + 4 \, B a^{3} d^{3} e + 6 \, A a^{3} d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d^{4} + 4 \, A a^{3} d^{3} e\right )} x^{2} \]
1/12*B*c^3*e^4*x^12 + 1/11*(4*B*c^3*d*e^3 + A*c^3*e^4)*x^11 + 1/10*(6*B*c^ 3*d^2*e^2 + 4*A*c^3*d*e^3 + 3*B*a*c^2*e^4)*x^10 + 1/9*(4*B*c^3*d^3*e + 6*A *c^3*d^2*e^2 + 12*B*a*c^2*d*e^3 + 3*A*a*c^2*e^4)*x^9 + A*a^3*d^4*x + 1/8*( B*c^3*d^4 + 4*A*c^3*d^3*e + 18*B*a*c^2*d^2*e^2 + 12*A*a*c^2*d*e^3 + 3*B*a^ 2*c*e^4)*x^8 + 1/7*(A*c^3*d^4 + 12*B*a*c^2*d^3*e + 18*A*a*c^2*d^2*e^2 + 12 *B*a^2*c*d*e^3 + 3*A*a^2*c*e^4)*x^7 + 1/6*(3*B*a*c^2*d^4 + 12*A*a*c^2*d^3* e + 18*B*a^2*c*d^2*e^2 + 12*A*a^2*c*d*e^3 + B*a^3*e^4)*x^6 + 1/5*(3*A*a*c^ 2*d^4 + 12*B*a^2*c*d^3*e + 18*A*a^2*c*d^2*e^2 + 4*B*a^3*d*e^3 + A*a^3*e^4) *x^5 + 1/4*(3*B*a^2*c*d^4 + 12*A*a^2*c*d^3*e + 6*B*a^3*d^2*e^2 + 4*A*a^3*d *e^3)*x^4 + 1/3*(3*A*a^2*c*d^4 + 4*B*a^3*d^3*e + 6*A*a^3*d^2*e^2)*x^3 + 1/ 2*(B*a^3*d^4 + 4*A*a^3*d^3*e)*x^2
Time = 0.05 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.69 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^3 \, dx=A a^{3} d^{4} x + \frac {B c^{3} e^{4} x^{12}}{12} + x^{11} \left (\frac {A c^{3} e^{4}}{11} + \frac {4 B c^{3} d e^{3}}{11}\right ) + x^{10} \cdot \left (\frac {2 A c^{3} d e^{3}}{5} + \frac {3 B a c^{2} e^{4}}{10} + \frac {3 B c^{3} d^{2} e^{2}}{5}\right ) + x^{9} \left (\frac {A a c^{2} e^{4}}{3} + \frac {2 A c^{3} d^{2} e^{2}}{3} + \frac {4 B a c^{2} d e^{3}}{3} + \frac {4 B c^{3} d^{3} e}{9}\right ) + x^{8} \cdot \left (\frac {3 A a c^{2} d e^{3}}{2} + \frac {A c^{3} d^{3} e}{2} + \frac {3 B a^{2} c e^{4}}{8} + \frac {9 B a c^{2} d^{2} e^{2}}{4} + \frac {B c^{3} d^{4}}{8}\right ) + x^{7} \cdot \left (\frac {3 A a^{2} c e^{4}}{7} + \frac {18 A a c^{2} d^{2} e^{2}}{7} + \frac {A c^{3} d^{4}}{7} + \frac {12 B a^{2} c d e^{3}}{7} + \frac {12 B a c^{2} d^{3} e}{7}\right ) + x^{6} \cdot \left (2 A a^{2} c d e^{3} + 2 A a c^{2} d^{3} e + \frac {B a^{3} e^{4}}{6} + 3 B a^{2} c d^{2} e^{2} + \frac {B a c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac {A a^{3} e^{4}}{5} + \frac {18 A a^{2} c d^{2} e^{2}}{5} + \frac {3 A a c^{2} d^{4}}{5} + \frac {4 B a^{3} d e^{3}}{5} + \frac {12 B a^{2} c d^{3} e}{5}\right ) + x^{4} \left (A a^{3} d e^{3} + 3 A a^{2} c d^{3} e + \frac {3 B a^{3} d^{2} e^{2}}{2} + \frac {3 B a^{2} c d^{4}}{4}\right ) + x^{3} \cdot \left (2 A a^{3} d^{2} e^{2} + A a^{2} c d^{4} + \frac {4 B a^{3} d^{3} e}{3}\right ) + x^{2} \cdot \left (2 A a^{3} d^{3} e + \frac {B a^{3} d^{4}}{2}\right ) \]
A*a**3*d**4*x + B*c**3*e**4*x**12/12 + x**11*(A*c**3*e**4/11 + 4*B*c**3*d* e**3/11) + x**10*(2*A*c**3*d*e**3/5 + 3*B*a*c**2*e**4/10 + 3*B*c**3*d**2*e **2/5) + x**9*(A*a*c**2*e**4/3 + 2*A*c**3*d**2*e**2/3 + 4*B*a*c**2*d*e**3/ 3 + 4*B*c**3*d**3*e/9) + x**8*(3*A*a*c**2*d*e**3/2 + A*c**3*d**3*e/2 + 3*B *a**2*c*e**4/8 + 9*B*a*c**2*d**2*e**2/4 + B*c**3*d**4/8) + x**7*(3*A*a**2* c*e**4/7 + 18*A*a*c**2*d**2*e**2/7 + A*c**3*d**4/7 + 12*B*a**2*c*d*e**3/7 + 12*B*a*c**2*d**3*e/7) + x**6*(2*A*a**2*c*d*e**3 + 2*A*a*c**2*d**3*e + B* a**3*e**4/6 + 3*B*a**2*c*d**2*e**2 + B*a*c**2*d**4/2) + x**5*(A*a**3*e**4/ 5 + 18*A*a**2*c*d**2*e**2/5 + 3*A*a*c**2*d**4/5 + 4*B*a**3*d*e**3/5 + 12*B *a**2*c*d**3*e/5) + x**4*(A*a**3*d*e**3 + 3*A*a**2*c*d**3*e + 3*B*a**3*d** 2*e**2/2 + 3*B*a**2*c*d**4/4) + x**3*(2*A*a**3*d**2*e**2 + A*a**2*c*d**4 + 4*B*a**3*d**3*e/3) + x**2*(2*A*a**3*d**3*e + B*a**3*d**4/2)
Time = 0.20 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.43 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^3 \, dx=\frac {1}{12} \, B c^{3} e^{4} x^{12} + \frac {1}{11} \, {\left (4 \, B c^{3} d e^{3} + A c^{3} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (6 \, B c^{3} d^{2} e^{2} + 4 \, A c^{3} d e^{3} + 3 \, B a c^{2} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{3} d^{3} e + 6 \, A c^{3} d^{2} e^{2} + 12 \, B a c^{2} d e^{3} + 3 \, A a c^{2} e^{4}\right )} x^{9} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (B c^{3} d^{4} + 4 \, A c^{3} d^{3} e + 18 \, B a c^{2} d^{2} e^{2} + 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (A c^{3} d^{4} + 12 \, B a c^{2} d^{3} e + 18 \, A a c^{2} d^{2} e^{2} + 12 \, B a^{2} c d e^{3} + 3 \, A a^{2} c e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, B a c^{2} d^{4} + 12 \, A a c^{2} d^{3} e + 18 \, B a^{2} c d^{2} e^{2} + 12 \, A a^{2} c d e^{3} + B a^{3} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, A a c^{2} d^{4} + 12 \, B a^{2} c d^{3} e + 18 \, A a^{2} c d^{2} e^{2} + 4 \, B a^{3} d e^{3} + A a^{3} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{4} + 12 \, A a^{2} c d^{3} e + 6 \, B a^{3} d^{2} e^{2} + 4 \, A a^{3} d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} c d^{4} + 4 \, B a^{3} d^{3} e + 6 \, A a^{3} d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d^{4} + 4 \, A a^{3} d^{3} e\right )} x^{2} \]
1/12*B*c^3*e^4*x^12 + 1/11*(4*B*c^3*d*e^3 + A*c^3*e^4)*x^11 + 1/10*(6*B*c^ 3*d^2*e^2 + 4*A*c^3*d*e^3 + 3*B*a*c^2*e^4)*x^10 + 1/9*(4*B*c^3*d^3*e + 6*A *c^3*d^2*e^2 + 12*B*a*c^2*d*e^3 + 3*A*a*c^2*e^4)*x^9 + A*a^3*d^4*x + 1/8*( B*c^3*d^4 + 4*A*c^3*d^3*e + 18*B*a*c^2*d^2*e^2 + 12*A*a*c^2*d*e^3 + 3*B*a^ 2*c*e^4)*x^8 + 1/7*(A*c^3*d^4 + 12*B*a*c^2*d^3*e + 18*A*a*c^2*d^2*e^2 + 12 *B*a^2*c*d*e^3 + 3*A*a^2*c*e^4)*x^7 + 1/6*(3*B*a*c^2*d^4 + 12*A*a*c^2*d^3* e + 18*B*a^2*c*d^2*e^2 + 12*A*a^2*c*d*e^3 + B*a^3*e^4)*x^6 + 1/5*(3*A*a*c^ 2*d^4 + 12*B*a^2*c*d^3*e + 18*A*a^2*c*d^2*e^2 + 4*B*a^3*d*e^3 + A*a^3*e^4) *x^5 + 1/4*(3*B*a^2*c*d^4 + 12*A*a^2*c*d^3*e + 6*B*a^3*d^2*e^2 + 4*A*a^3*d *e^3)*x^4 + 1/3*(3*A*a^2*c*d^4 + 4*B*a^3*d^3*e + 6*A*a^3*d^2*e^2)*x^3 + 1/ 2*(B*a^3*d^4 + 4*A*a^3*d^3*e)*x^2
Time = 0.27 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.60 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^3 \, dx=\frac {1}{12} \, B c^{3} e^{4} x^{12} + \frac {4}{11} \, B c^{3} d e^{3} x^{11} + \frac {1}{11} \, A c^{3} e^{4} x^{11} + \frac {3}{5} \, B c^{3} d^{2} e^{2} x^{10} + \frac {2}{5} \, A c^{3} d e^{3} x^{10} + \frac {3}{10} \, B a c^{2} e^{4} x^{10} + \frac {4}{9} \, B c^{3} d^{3} e x^{9} + \frac {2}{3} \, A c^{3} d^{2} e^{2} x^{9} + \frac {4}{3} \, B a c^{2} d e^{3} x^{9} + \frac {1}{3} \, A a c^{2} e^{4} x^{9} + \frac {1}{8} \, B c^{3} d^{4} x^{8} + \frac {1}{2} \, A c^{3} d^{3} e x^{8} + \frac {9}{4} \, B a c^{2} d^{2} e^{2} x^{8} + \frac {3}{2} \, A a c^{2} d e^{3} x^{8} + \frac {3}{8} \, B a^{2} c e^{4} x^{8} + \frac {1}{7} \, A c^{3} d^{4} x^{7} + \frac {12}{7} \, B a c^{2} d^{3} e x^{7} + \frac {18}{7} \, A a c^{2} d^{2} e^{2} x^{7} + \frac {12}{7} \, B a^{2} c d e^{3} x^{7} + \frac {3}{7} \, A a^{2} c e^{4} x^{7} + \frac {1}{2} \, B a c^{2} d^{4} x^{6} + 2 \, A a c^{2} d^{3} e x^{6} + 3 \, B a^{2} c d^{2} e^{2} x^{6} + 2 \, A a^{2} c d e^{3} x^{6} + \frac {1}{6} \, B a^{3} e^{4} x^{6} + \frac {3}{5} \, A a c^{2} d^{4} x^{5} + \frac {12}{5} \, B a^{2} c d^{3} e x^{5} + \frac {18}{5} \, A a^{2} c d^{2} e^{2} x^{5} + \frac {4}{5} \, B a^{3} d e^{3} x^{5} + \frac {1}{5} \, A a^{3} e^{4} x^{5} + \frac {3}{4} \, B a^{2} c d^{4} x^{4} + 3 \, A a^{2} c d^{3} e x^{4} + \frac {3}{2} \, B a^{3} d^{2} e^{2} x^{4} + A a^{3} d e^{3} x^{4} + A a^{2} c d^{4} x^{3} + \frac {4}{3} \, B a^{3} d^{3} e x^{3} + 2 \, A a^{3} d^{2} e^{2} x^{3} + \frac {1}{2} \, B a^{3} d^{4} x^{2} + 2 \, A a^{3} d^{3} e x^{2} + A a^{3} d^{4} x \]
1/12*B*c^3*e^4*x^12 + 4/11*B*c^3*d*e^3*x^11 + 1/11*A*c^3*e^4*x^11 + 3/5*B* c^3*d^2*e^2*x^10 + 2/5*A*c^3*d*e^3*x^10 + 3/10*B*a*c^2*e^4*x^10 + 4/9*B*c^ 3*d^3*e*x^9 + 2/3*A*c^3*d^2*e^2*x^9 + 4/3*B*a*c^2*d*e^3*x^9 + 1/3*A*a*c^2* e^4*x^9 + 1/8*B*c^3*d^4*x^8 + 1/2*A*c^3*d^3*e*x^8 + 9/4*B*a*c^2*d^2*e^2*x^ 8 + 3/2*A*a*c^2*d*e^3*x^8 + 3/8*B*a^2*c*e^4*x^8 + 1/7*A*c^3*d^4*x^7 + 12/7 *B*a*c^2*d^3*e*x^7 + 18/7*A*a*c^2*d^2*e^2*x^7 + 12/7*B*a^2*c*d*e^3*x^7 + 3 /7*A*a^2*c*e^4*x^7 + 1/2*B*a*c^2*d^4*x^6 + 2*A*a*c^2*d^3*e*x^6 + 3*B*a^2*c *d^2*e^2*x^6 + 2*A*a^2*c*d*e^3*x^6 + 1/6*B*a^3*e^4*x^6 + 3/5*A*a*c^2*d^4*x ^5 + 12/5*B*a^2*c*d^3*e*x^5 + 18/5*A*a^2*c*d^2*e^2*x^5 + 4/5*B*a^3*d*e^3*x ^5 + 1/5*A*a^3*e^4*x^5 + 3/4*B*a^2*c*d^4*x^4 + 3*A*a^2*c*d^3*e*x^4 + 3/2*B *a^3*d^2*e^2*x^4 + A*a^3*d*e^3*x^4 + A*a^2*c*d^4*x^3 + 4/3*B*a^3*d^3*e*x^3 + 2*A*a^3*d^2*e^2*x^3 + 1/2*B*a^3*d^4*x^2 + 2*A*a^3*d^3*e*x^2 + A*a^3*d^4 *x
Time = 0.22 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.31 \[ \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^3 \, dx=x^5\,\left (\frac {4\,B\,a^3\,d\,e^3}{5}+\frac {A\,a^3\,e^4}{5}+\frac {12\,B\,a^2\,c\,d^3\,e}{5}+\frac {18\,A\,a^2\,c\,d^2\,e^2}{5}+\frac {3\,A\,a\,c^2\,d^4}{5}\right )+x^8\,\left (\frac {3\,B\,a^2\,c\,e^4}{8}+\frac {9\,B\,a\,c^2\,d^2\,e^2}{4}+\frac {3\,A\,a\,c^2\,d\,e^3}{2}+\frac {B\,c^3\,d^4}{8}+\frac {A\,c^3\,d^3\,e}{2}\right )+x^6\,\left (\frac {B\,a^3\,e^4}{6}+3\,B\,a^2\,c\,d^2\,e^2+2\,A\,a^2\,c\,d\,e^3+\frac {B\,a\,c^2\,d^4}{2}+2\,A\,a\,c^2\,d^3\,e\right )+x^7\,\left (\frac {12\,B\,a^2\,c\,d\,e^3}{7}+\frac {3\,A\,a^2\,c\,e^4}{7}+\frac {12\,B\,a\,c^2\,d^3\,e}{7}+\frac {18\,A\,a\,c^2\,d^2\,e^2}{7}+\frac {A\,c^3\,d^4}{7}\right )+\frac {a^3\,d^3\,x^2\,\left (4\,A\,e+B\,d\right )}{2}+\frac {c^3\,e^3\,x^{11}\,\left (A\,e+4\,B\,d\right )}{11}+\frac {a^2\,d^2\,x^3\,\left (3\,A\,c\,d^2+4\,B\,a\,d\,e+6\,A\,a\,e^2\right )}{3}+\frac {c^2\,e^2\,x^{10}\,\left (6\,B\,c\,d^2+4\,A\,c\,d\,e+3\,B\,a\,e^2\right )}{10}+A\,a^3\,d^4\,x+\frac {a^2\,d\,x^4\,\left (3\,B\,c\,d^3+12\,A\,c\,d^2\,e+6\,B\,a\,d\,e^2+4\,A\,a\,e^3\right )}{4}+\frac {c^2\,e\,x^9\,\left (4\,B\,c\,d^3+6\,A\,c\,d^2\,e+12\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{9}+\frac {B\,c^3\,e^4\,x^{12}}{12} \]
x^5*((A*a^3*e^4)/5 + (3*A*a*c^2*d^4)/5 + (4*B*a^3*d*e^3)/5 + (18*A*a^2*c*d ^2*e^2)/5 + (12*B*a^2*c*d^3*e)/5) + x^8*((B*c^3*d^4)/8 + (3*B*a^2*c*e^4)/8 + (A*c^3*d^3*e)/2 + (9*B*a*c^2*d^2*e^2)/4 + (3*A*a*c^2*d*e^3)/2) + x^6*(( B*a^3*e^4)/6 + (B*a*c^2*d^4)/2 + 3*B*a^2*c*d^2*e^2 + 2*A*a*c^2*d^3*e + 2*A *a^2*c*d*e^3) + x^7*((A*c^3*d^4)/7 + (3*A*a^2*c*e^4)/7 + (18*A*a*c^2*d^2*e ^2)/7 + (12*B*a*c^2*d^3*e)/7 + (12*B*a^2*c*d*e^3)/7) + (a^3*d^3*x^2*(4*A*e + B*d))/2 + (c^3*e^3*x^11*(A*e + 4*B*d))/11 + (a^2*d^2*x^3*(6*A*a*e^2 + 3 *A*c*d^2 + 4*B*a*d*e))/3 + (c^2*e^2*x^10*(3*B*a*e^2 + 6*B*c*d^2 + 4*A*c*d* e))/10 + A*a^3*d^4*x + (a^2*d*x^4*(4*A*a*e^3 + 3*B*c*d^3 + 6*B*a*d*e^2 + 1 2*A*c*d^2*e))/4 + (c^2*e*x^9*(3*A*a*e^3 + 4*B*c*d^3 + 12*B*a*d*e^2 + 6*A*c *d^2*e))/9 + (B*c^3*e^4*x^12)/12