Integrand size = 17, antiderivative size = 278 \[ \int (d+e x)^5 \left (a+c x^2\right )^4 \, dx=\frac {\left (c d^2+a e^2\right )^4 (d+e x)^6}{6 e^9}-\frac {8 c d \left (c d^2+a e^2\right )^3 (d+e x)^7}{7 e^9}+\frac {c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^8}{2 e^9}-\frac {8 c^2 d \left (c d^2+a e^2\right ) \left (7 c d^2+3 a e^2\right ) (d+e x)^9}{9 e^9}+\frac {c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^{10}}{5 e^9}-\frac {8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^{11}}{11 e^9}+\frac {c^3 \left (7 c d^2+a e^2\right ) (d+e x)^{12}}{3 e^9}-\frac {8 c^4 d (d+e x)^{13}}{13 e^9}+\frac {c^4 (d+e x)^{14}}{14 e^9} \]
1/6*(a*e^2+c*d^2)^4*(e*x+d)^6/e^9-8/7*c*d*(a*e^2+c*d^2)^3*(e*x+d)^7/e^9+1/ 2*c*(a*e^2+c*d^2)^2*(a*e^2+7*c*d^2)*(e*x+d)^8/e^9-8/9*c^2*d*(a*e^2+c*d^2)* (3*a*e^2+7*c*d^2)*(e*x+d)^9/e^9+1/5*c^2*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2*d ^4)*(e*x+d)^10/e^9-8/11*c^3*d*(3*a*e^2+7*c*d^2)*(e*x+d)^11/e^9+1/3*c^3*(a* e^2+7*c*d^2)*(e*x+d)^12/e^9-8/13*c^4*d*(e*x+d)^13/e^9+1/14*c^4*(e*x+d)^14/ e^9
Time = 0.10 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.10 \[ \int (d+e x)^5 \left (a+c x^2\right )^4 \, dx=\frac {x \left (15015 a^4 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+2145 a^3 c x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+429 a^2 c^2 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+65 a c^3 x^6 \left (792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right )+5 c^4 x^8 \left (2002 d^5+9009 d^4 e x+16380 d^3 e^2 x^2+15015 d^2 e^3 x^3+6930 d e^4 x^4+1287 e^5 x^5\right )\right )}{90090} \]
(x*(15015*a^4*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d* e^4*x^4 + e^5*x^5) + 2145*a^3*c*x^2*(56*d^5 + 210*d^4*e*x + 336*d^3*e^2*x^ 2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + 429*a^2*c^2*x^4*(252*d ^5 + 1050*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5) + 65*a*c^3*x^6*(792*d^5 + 3465*d^4*e*x + 6160*d^3*e^2*x^2 + 5 544*d^2*e^3*x^3 + 2520*d*e^4*x^4 + 462*e^5*x^5) + 5*c^4*x^8*(2002*d^5 + 90 09*d^4*e*x + 16380*d^3*e^2*x^2 + 15015*d^2*e^3*x^3 + 6930*d*e^4*x^4 + 1287 *e^5*x^5)))/90090
Time = 0.64 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {476, 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.
| \(\displaystyle \int \left (a+c x^2\right )^4 (d+e x)^5 \, dx\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \int \left (\frac {2 c^2 (d+e x)^9 \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{e^8}+\frac {4 c^3 (d+e x)^{11} \left (a e^2+7 c d^2\right )}{e^8}-\frac {8 c^3 d (d+e x)^{10} \left (3 a e^2+7 c d^2\right )}{e^8}+\frac {8 c^2 d (d+e x)^8 \left (-3 a e^2-7 c d^2\right ) \left (a e^2+c d^2\right )}{e^8}+\frac {4 c (d+e x)^7 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{e^8}-\frac {8 c d (d+e x)^6 \left (a e^2+c d^2\right )^3}{e^8}+\frac {(d+e x)^5 \left (a e^2+c d^2\right )^4}{e^8}+\frac {c^4 (d+e x)^{13}}{e^8}-\frac {8 c^4 d (d+e x)^{12}}{e^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^2 (d+e x)^{10} \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{5 e^9}+\frac {c^3 (d+e x)^{12} \left (a e^2+7 c d^2\right )}{3 e^9}-\frac {8 c^3 d (d+e x)^{11} \left (3 a e^2+7 c d^2\right )}{11 e^9}-\frac {8 c^2 d (d+e x)^9 \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{9 e^9}+\frac {c (d+e x)^8 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{2 e^9}-\frac {8 c d (d+e x)^7 \left (a e^2+c d^2\right )^3}{7 e^9}+\frac {(d+e x)^6 \left (a e^2+c d^2\right )^4}{6 e^9}+\frac {c^4 (d+e x)^{14}}{14 e^9}-\frac {8 c^4 d (d+e x)^{13}}{13 e^9}\) |
((c*d^2 + a*e^2)^4*(d + e*x)^6)/(6*e^9) - (8*c*d*(c*d^2 + a*e^2)^3*(d + e* x)^7)/(7*e^9) + (c*(c*d^2 + a*e^2)^2*(7*c*d^2 + a*e^2)*(d + e*x)^8)/(2*e^9 ) - (8*c^2*d*(c*d^2 + a*e^2)*(7*c*d^2 + 3*a*e^2)*(d + e*x)^9)/(9*e^9) + (c ^2*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)*(d + e*x)^10)/(5*e^9) - (8*c^ 3*d*(7*c*d^2 + 3*a*e^2)*(d + e*x)^11)/(11*e^9) + (c^3*(7*c*d^2 + a*e^2)*(d + e*x)^12)/(3*e^9) - (8*c^4*d*(d + e*x)^13)/(13*e^9) + (c^4*(d + e*x)^14) /(14*e^9)
3.5.90.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Time = 2.27 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.33
| method | result | size |
| norman | \(a^{4} d^{5} x +\frac {5 d^{4} e \,a^{4} x^{2}}{2}+\left (\frac {10}{3} d^{3} e^{2} a^{4}+\frac {4}{3} d^{5} c \,a^{3}\right ) x^{3}+\left (\frac {5}{2} d^{2} e^{3} a^{4}+5 d^{4} e c \,a^{3}\right ) x^{4}+\left (d \,e^{4} a^{4}+8 d^{3} e^{2} c \,a^{3}+\frac {6}{5} d^{5} a^{2} c^{2}\right ) x^{5}+\left (\frac {1}{6} e^{5} a^{4}+\frac {20}{3} d^{2} e^{3} c \,a^{3}+5 d^{4} e \,a^{2} c^{2}\right ) x^{6}+\left (\frac {20}{7} d \,e^{4} c \,a^{3}+\frac {60}{7} d^{3} e^{2} a^{2} c^{2}+\frac {4}{7} d^{5} c^{3} a \right ) x^{7}+\left (\frac {1}{2} e^{5} c \,a^{3}+\frac {15}{2} d^{2} e^{3} a^{2} c^{2}+\frac {5}{2} d^{4} e \,c^{3} a \right ) x^{8}+\left (\frac {10}{3} d \,e^{4} a^{2} c^{2}+\frac {40}{9} d^{3} e^{2} c^{3} a +\frac {1}{9} c^{4} d^{5}\right ) x^{9}+\left (\frac {3}{5} e^{5} a^{2} c^{2}+4 d^{2} e^{3} c^{3} a +\frac {1}{2} d^{4} e \,c^{4}\right ) x^{10}+\left (\frac {20}{11} d \,e^{4} c^{3} a +\frac {10}{11} d^{3} e^{2} c^{4}\right ) x^{11}+\left (\frac {1}{3} e^{5} c^{3} a +\frac {5}{6} d^{2} e^{3} c^{4}\right ) x^{12}+\frac {5 d \,e^{4} c^{4} x^{13}}{13}+\frac {e^{5} c^{4} x^{14}}{14}\) | \(370\) |
| default | \(\frac {e^{5} c^{4} x^{14}}{14}+\frac {5 d \,e^{4} c^{4} x^{13}}{13}+\frac {\left (4 e^{5} c^{3} a +10 d^{2} e^{3} c^{4}\right ) x^{12}}{12}+\frac {\left (20 d \,e^{4} c^{3} a +10 d^{3} e^{2} c^{4}\right ) x^{11}}{11}+\frac {\left (6 e^{5} a^{2} c^{2}+40 d^{2} e^{3} c^{3} a +5 d^{4} e \,c^{4}\right ) x^{10}}{10}+\frac {\left (30 d \,e^{4} a^{2} c^{2}+40 d^{3} e^{2} c^{3} a +c^{4} d^{5}\right ) x^{9}}{9}+\frac {\left (4 e^{5} c \,a^{3}+60 d^{2} e^{3} a^{2} c^{2}+20 d^{4} e \,c^{3} a \right ) x^{8}}{8}+\frac {\left (20 d \,e^{4} c \,a^{3}+60 d^{3} e^{2} a^{2} c^{2}+4 d^{5} c^{3} a \right ) x^{7}}{7}+\frac {\left (e^{5} a^{4}+40 d^{2} e^{3} c \,a^{3}+30 d^{4} e \,a^{2} c^{2}\right ) x^{6}}{6}+\frac {\left (5 d \,e^{4} a^{4}+40 d^{3} e^{2} c \,a^{3}+6 d^{5} a^{2} c^{2}\right ) x^{5}}{5}+\frac {\left (10 d^{2} e^{3} a^{4}+20 d^{4} e c \,a^{3}\right ) x^{4}}{4}+\frac {\left (10 d^{3} e^{2} a^{4}+4 d^{5} c \,a^{3}\right ) x^{3}}{3}+\frac {5 d^{4} e \,a^{4} x^{2}}{2}+a^{4} d^{5} x\) | \(379\) |
| gosper | \(\frac {5}{2} d^{4} e \,a^{4} x^{2}+5 x^{4} d^{4} e c \,a^{3}+\frac {15}{2} x^{8} d^{2} e^{3} a^{2} c^{2}+5 x^{6} d^{4} e \,a^{2} c^{2}+\frac {1}{2} x^{8} e^{5} c \,a^{3}+\frac {3}{5} x^{10} e^{5} a^{2} c^{2}+\frac {1}{2} x^{10} d^{4} e \,c^{4}+\frac {10}{11} x^{11} d^{3} e^{2} c^{4}+\frac {60}{7} x^{7} d^{3} e^{2} a^{2} c^{2}+\frac {5}{2} x^{8} d^{4} e \,c^{3} a +\frac {10}{3} x^{9} d \,e^{4} a^{2} c^{2}+\frac {40}{9} x^{9} d^{3} e^{2} c^{3} a +4 x^{10} d^{2} e^{3} c^{3} a +\frac {20}{11} x^{11} d \,e^{4} c^{3} a +8 x^{5} d^{3} e^{2} c \,a^{3}+\frac {20}{3} x^{6} d^{2} e^{3} c \,a^{3}+\frac {20}{7} x^{7} d \,e^{4} c \,a^{3}+\frac {10}{3} x^{3} d^{3} e^{2} a^{4}+\frac {4}{3} x^{3} d^{5} c \,a^{3}+\frac {5}{13} d \,e^{4} c^{4} x^{13}+\frac {1}{6} x^{6} e^{5} a^{4}+\frac {1}{9} x^{9} c^{4} d^{5}+\frac {1}{3} x^{12} e^{5} c^{3} a +\frac {5}{6} x^{12} d^{2} e^{3} c^{4}+\frac {5}{2} x^{4} d^{2} e^{3} a^{4}+x^{5} d \,e^{4} a^{4}+\frac {6}{5} x^{5} d^{5} a^{2} c^{2}+\frac {4}{7} x^{7} d^{5} c^{3} a +a^{4} d^{5} x +\frac {1}{14} e^{5} c^{4} x^{14}\) | \(398\) |
| risch | \(\frac {5}{2} d^{4} e \,a^{4} x^{2}+5 x^{4} d^{4} e c \,a^{3}+\frac {15}{2} x^{8} d^{2} e^{3} a^{2} c^{2}+5 x^{6} d^{4} e \,a^{2} c^{2}+\frac {1}{2} x^{8} e^{5} c \,a^{3}+\frac {3}{5} x^{10} e^{5} a^{2} c^{2}+\frac {1}{2} x^{10} d^{4} e \,c^{4}+\frac {10}{11} x^{11} d^{3} e^{2} c^{4}+\frac {60}{7} x^{7} d^{3} e^{2} a^{2} c^{2}+\frac {5}{2} x^{8} d^{4} e \,c^{3} a +\frac {10}{3} x^{9} d \,e^{4} a^{2} c^{2}+\frac {40}{9} x^{9} d^{3} e^{2} c^{3} a +4 x^{10} d^{2} e^{3} c^{3} a +\frac {20}{11} x^{11} d \,e^{4} c^{3} a +8 x^{5} d^{3} e^{2} c \,a^{3}+\frac {20}{3} x^{6} d^{2} e^{3} c \,a^{3}+\frac {20}{7} x^{7} d \,e^{4} c \,a^{3}+\frac {10}{3} x^{3} d^{3} e^{2} a^{4}+\frac {4}{3} x^{3} d^{5} c \,a^{3}+\frac {5}{13} d \,e^{4} c^{4} x^{13}+\frac {1}{6} x^{6} e^{5} a^{4}+\frac {1}{9} x^{9} c^{4} d^{5}+\frac {1}{3} x^{12} e^{5} c^{3} a +\frac {5}{6} x^{12} d^{2} e^{3} c^{4}+\frac {5}{2} x^{4} d^{2} e^{3} a^{4}+x^{5} d \,e^{4} a^{4}+\frac {6}{5} x^{5} d^{5} a^{2} c^{2}+\frac {4}{7} x^{7} d^{5} c^{3} a +a^{4} d^{5} x +\frac {1}{14} e^{5} c^{4} x^{14}\) | \(398\) |
| parallelrisch | \(\frac {5}{2} d^{4} e \,a^{4} x^{2}+5 x^{4} d^{4} e c \,a^{3}+\frac {15}{2} x^{8} d^{2} e^{3} a^{2} c^{2}+5 x^{6} d^{4} e \,a^{2} c^{2}+\frac {1}{2} x^{8} e^{5} c \,a^{3}+\frac {3}{5} x^{10} e^{5} a^{2} c^{2}+\frac {1}{2} x^{10} d^{4} e \,c^{4}+\frac {10}{11} x^{11} d^{3} e^{2} c^{4}+\frac {60}{7} x^{7} d^{3} e^{2} a^{2} c^{2}+\frac {5}{2} x^{8} d^{4} e \,c^{3} a +\frac {10}{3} x^{9} d \,e^{4} a^{2} c^{2}+\frac {40}{9} x^{9} d^{3} e^{2} c^{3} a +4 x^{10} d^{2} e^{3} c^{3} a +\frac {20}{11} x^{11} d \,e^{4} c^{3} a +8 x^{5} d^{3} e^{2} c \,a^{3}+\frac {20}{3} x^{6} d^{2} e^{3} c \,a^{3}+\frac {20}{7} x^{7} d \,e^{4} c \,a^{3}+\frac {10}{3} x^{3} d^{3} e^{2} a^{4}+\frac {4}{3} x^{3} d^{5} c \,a^{3}+\frac {5}{13} d \,e^{4} c^{4} x^{13}+\frac {1}{6} x^{6} e^{5} a^{4}+\frac {1}{9} x^{9} c^{4} d^{5}+\frac {1}{3} x^{12} e^{5} c^{3} a +\frac {5}{6} x^{12} d^{2} e^{3} c^{4}+\frac {5}{2} x^{4} d^{2} e^{3} a^{4}+x^{5} d \,e^{4} a^{4}+\frac {6}{5} x^{5} d^{5} a^{2} c^{2}+\frac {4}{7} x^{7} d^{5} c^{3} a +a^{4} d^{5} x +\frac {1}{14} e^{5} c^{4} x^{14}\) | \(398\) |
a^4*d^5*x+5/2*d^4*e*a^4*x^2+(10/3*d^3*e^2*a^4+4/3*d^5*c*a^3)*x^3+(5/2*d^2* e^3*a^4+5*d^4*e*c*a^3)*x^4+(d*e^4*a^4+8*d^3*e^2*c*a^3+6/5*d^5*a^2*c^2)*x^5 +(1/6*e^5*a^4+20/3*d^2*e^3*c*a^3+5*d^4*e*a^2*c^2)*x^6+(20/7*d*e^4*c*a^3+60 /7*d^3*e^2*a^2*c^2+4/7*d^5*c^3*a)*x^7+(1/2*e^5*c*a^3+15/2*d^2*e^3*a^2*c^2+ 5/2*d^4*e*c^3*a)*x^8+(10/3*d*e^4*a^2*c^2+40/9*d^3*e^2*c^3*a+1/9*c^4*d^5)*x ^9+(3/5*e^5*a^2*c^2+4*d^2*e^3*c^3*a+1/2*d^4*e*c^4)*x^10+(20/11*d*e^4*c^3*a +10/11*d^3*e^2*c^4)*x^11+(1/3*e^5*c^3*a+5/6*d^2*e^3*c^4)*x^12+5/13*d*e^4*c ^4*x^13+1/14*e^5*c^4*x^14
Time = 0.28 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.35 \[ \int (d+e x)^5 \left (a+c x^2\right )^4 \, dx=\frac {1}{14} \, c^{4} e^{5} x^{14} + \frac {5}{13} \, c^{4} d e^{4} x^{13} + \frac {1}{6} \, {\left (5 \, c^{4} d^{2} e^{3} + 2 \, a c^{3} e^{5}\right )} x^{12} + \frac {10}{11} \, {\left (c^{4} d^{3} e^{2} + 2 \, a c^{3} d e^{4}\right )} x^{11} + \frac {5}{2} \, a^{4} d^{4} e x^{2} + \frac {1}{10} \, {\left (5 \, c^{4} d^{4} e + 40 \, a c^{3} d^{2} e^{3} + 6 \, a^{2} c^{2} e^{5}\right )} x^{10} + a^{4} d^{5} x + \frac {1}{9} \, {\left (c^{4} d^{5} + 40 \, a c^{3} d^{3} e^{2} + 30 \, a^{2} c^{2} d e^{4}\right )} x^{9} + \frac {1}{2} \, {\left (5 \, a c^{3} d^{4} e + 15 \, a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{8} + \frac {4}{7} \, {\left (a c^{3} d^{5} + 15 \, a^{2} c^{2} d^{3} e^{2} + 5 \, a^{3} c d e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (30 \, a^{2} c^{2} d^{4} e + 40 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, a^{2} c^{2} d^{5} + 40 \, a^{3} c d^{3} e^{2} + 5 \, a^{4} d e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (2 \, a^{3} c d^{4} e + a^{4} d^{2} e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (2 \, a^{3} c d^{5} + 5 \, a^{4} d^{3} e^{2}\right )} x^{3} \]
1/14*c^4*e^5*x^14 + 5/13*c^4*d*e^4*x^13 + 1/6*(5*c^4*d^2*e^3 + 2*a*c^3*e^5 )*x^12 + 10/11*(c^4*d^3*e^2 + 2*a*c^3*d*e^4)*x^11 + 5/2*a^4*d^4*e*x^2 + 1/ 10*(5*c^4*d^4*e + 40*a*c^3*d^2*e^3 + 6*a^2*c^2*e^5)*x^10 + a^4*d^5*x + 1/9 *(c^4*d^5 + 40*a*c^3*d^3*e^2 + 30*a^2*c^2*d*e^4)*x^9 + 1/2*(5*a*c^3*d^4*e + 15*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^8 + 4/7*(a*c^3*d^5 + 15*a^2*c^2*d^3*e^ 2 + 5*a^3*c*d*e^4)*x^7 + 1/6*(30*a^2*c^2*d^4*e + 40*a^3*c*d^2*e^3 + a^4*e^ 5)*x^6 + 1/5*(6*a^2*c^2*d^5 + 40*a^3*c*d^3*e^2 + 5*a^4*d*e^4)*x^5 + 5/2*(2 *a^3*c*d^4*e + a^4*d^2*e^3)*x^4 + 2/3*(2*a^3*c*d^5 + 5*a^4*d^3*e^2)*x^3
Time = 0.05 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.50 \[ \int (d+e x)^5 \left (a+c x^2\right )^4 \, dx=a^{4} d^{5} x + \frac {5 a^{4} d^{4} e x^{2}}{2} + \frac {5 c^{4} d e^{4} x^{13}}{13} + \frac {c^{4} e^{5} x^{14}}{14} + x^{12} \left (\frac {a c^{3} e^{5}}{3} + \frac {5 c^{4} d^{2} e^{3}}{6}\right ) + x^{11} \cdot \left (\frac {20 a c^{3} d e^{4}}{11} + \frac {10 c^{4} d^{3} e^{2}}{11}\right ) + x^{10} \cdot \left (\frac {3 a^{2} c^{2} e^{5}}{5} + 4 a c^{3} d^{2} e^{3} + \frac {c^{4} d^{4} e}{2}\right ) + x^{9} \cdot \left (\frac {10 a^{2} c^{2} d e^{4}}{3} + \frac {40 a c^{3} d^{3} e^{2}}{9} + \frac {c^{4} d^{5}}{9}\right ) + x^{8} \left (\frac {a^{3} c e^{5}}{2} + \frac {15 a^{2} c^{2} d^{2} e^{3}}{2} + \frac {5 a c^{3} d^{4} e}{2}\right ) + x^{7} \cdot \left (\frac {20 a^{3} c d e^{4}}{7} + \frac {60 a^{2} c^{2} d^{3} e^{2}}{7} + \frac {4 a c^{3} d^{5}}{7}\right ) + x^{6} \left (\frac {a^{4} e^{5}}{6} + \frac {20 a^{3} c d^{2} e^{3}}{3} + 5 a^{2} c^{2} d^{4} e\right ) + x^{5} \left (a^{4} d e^{4} + 8 a^{3} c d^{3} e^{2} + \frac {6 a^{2} c^{2} d^{5}}{5}\right ) + x^{4} \cdot \left (\frac {5 a^{4} d^{2} e^{3}}{2} + 5 a^{3} c d^{4} e\right ) + x^{3} \cdot \left (\frac {10 a^{4} d^{3} e^{2}}{3} + \frac {4 a^{3} c d^{5}}{3}\right ) \]
a**4*d**5*x + 5*a**4*d**4*e*x**2/2 + 5*c**4*d*e**4*x**13/13 + c**4*e**5*x* *14/14 + x**12*(a*c**3*e**5/3 + 5*c**4*d**2*e**3/6) + x**11*(20*a*c**3*d*e **4/11 + 10*c**4*d**3*e**2/11) + x**10*(3*a**2*c**2*e**5/5 + 4*a*c**3*d**2 *e**3 + c**4*d**4*e/2) + x**9*(10*a**2*c**2*d*e**4/3 + 40*a*c**3*d**3*e**2 /9 + c**4*d**5/9) + x**8*(a**3*c*e**5/2 + 15*a**2*c**2*d**2*e**3/2 + 5*a*c **3*d**4*e/2) + x**7*(20*a**3*c*d*e**4/7 + 60*a**2*c**2*d**3*e**2/7 + 4*a* c**3*d**5/7) + x**6*(a**4*e**5/6 + 20*a**3*c*d**2*e**3/3 + 5*a**2*c**2*d** 4*e) + x**5*(a**4*d*e**4 + 8*a**3*c*d**3*e**2 + 6*a**2*c**2*d**5/5) + x**4 *(5*a**4*d**2*e**3/2 + 5*a**3*c*d**4*e) + x**3*(10*a**4*d**3*e**2/3 + 4*a* *3*c*d**5/3)
Time = 0.20 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.35 \[ \int (d+e x)^5 \left (a+c x^2\right )^4 \, dx=\frac {1}{14} \, c^{4} e^{5} x^{14} + \frac {5}{13} \, c^{4} d e^{4} x^{13} + \frac {1}{6} \, {\left (5 \, c^{4} d^{2} e^{3} + 2 \, a c^{3} e^{5}\right )} x^{12} + \frac {10}{11} \, {\left (c^{4} d^{3} e^{2} + 2 \, a c^{3} d e^{4}\right )} x^{11} + \frac {5}{2} \, a^{4} d^{4} e x^{2} + \frac {1}{10} \, {\left (5 \, c^{4} d^{4} e + 40 \, a c^{3} d^{2} e^{3} + 6 \, a^{2} c^{2} e^{5}\right )} x^{10} + a^{4} d^{5} x + \frac {1}{9} \, {\left (c^{4} d^{5} + 40 \, a c^{3} d^{3} e^{2} + 30 \, a^{2} c^{2} d e^{4}\right )} x^{9} + \frac {1}{2} \, {\left (5 \, a c^{3} d^{4} e + 15 \, a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{8} + \frac {4}{7} \, {\left (a c^{3} d^{5} + 15 \, a^{2} c^{2} d^{3} e^{2} + 5 \, a^{3} c d e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (30 \, a^{2} c^{2} d^{4} e + 40 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, a^{2} c^{2} d^{5} + 40 \, a^{3} c d^{3} e^{2} + 5 \, a^{4} d e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (2 \, a^{3} c d^{4} e + a^{4} d^{2} e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (2 \, a^{3} c d^{5} + 5 \, a^{4} d^{3} e^{2}\right )} x^{3} \]
1/14*c^4*e^5*x^14 + 5/13*c^4*d*e^4*x^13 + 1/6*(5*c^4*d^2*e^3 + 2*a*c^3*e^5 )*x^12 + 10/11*(c^4*d^3*e^2 + 2*a*c^3*d*e^4)*x^11 + 5/2*a^4*d^4*e*x^2 + 1/ 10*(5*c^4*d^4*e + 40*a*c^3*d^2*e^3 + 6*a^2*c^2*e^5)*x^10 + a^4*d^5*x + 1/9 *(c^4*d^5 + 40*a*c^3*d^3*e^2 + 30*a^2*c^2*d*e^4)*x^9 + 1/2*(5*a*c^3*d^4*e + 15*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^8 + 4/7*(a*c^3*d^5 + 15*a^2*c^2*d^3*e^ 2 + 5*a^3*c*d*e^4)*x^7 + 1/6*(30*a^2*c^2*d^4*e + 40*a^3*c*d^2*e^3 + a^4*e^ 5)*x^6 + 1/5*(6*a^2*c^2*d^5 + 40*a^3*c*d^3*e^2 + 5*a^4*d*e^4)*x^5 + 5/2*(2 *a^3*c*d^4*e + a^4*d^2*e^3)*x^4 + 2/3*(2*a^3*c*d^5 + 5*a^4*d^3*e^2)*x^3
Time = 0.27 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.43 \[ \int (d+e x)^5 \left (a+c x^2\right )^4 \, dx=\frac {1}{14} \, c^{4} e^{5} x^{14} + \frac {5}{13} \, c^{4} d e^{4} x^{13} + \frac {5}{6} \, c^{4} d^{2} e^{3} x^{12} + \frac {1}{3} \, a c^{3} e^{5} x^{12} + \frac {10}{11} \, c^{4} d^{3} e^{2} x^{11} + \frac {20}{11} \, a c^{3} d e^{4} x^{11} + \frac {1}{2} \, c^{4} d^{4} e x^{10} + 4 \, a c^{3} d^{2} e^{3} x^{10} + \frac {3}{5} \, a^{2} c^{2} e^{5} x^{10} + \frac {1}{9} \, c^{4} d^{5} x^{9} + \frac {40}{9} \, a c^{3} d^{3} e^{2} x^{9} + \frac {10}{3} \, a^{2} c^{2} d e^{4} x^{9} + \frac {5}{2} \, a c^{3} d^{4} e x^{8} + \frac {15}{2} \, a^{2} c^{2} d^{2} e^{3} x^{8} + \frac {1}{2} \, a^{3} c e^{5} x^{8} + \frac {4}{7} \, a c^{3} d^{5} x^{7} + \frac {60}{7} \, a^{2} c^{2} d^{3} e^{2} x^{7} + \frac {20}{7} \, a^{3} c d e^{4} x^{7} + 5 \, a^{2} c^{2} d^{4} e x^{6} + \frac {20}{3} \, a^{3} c d^{2} e^{3} x^{6} + \frac {1}{6} \, a^{4} e^{5} x^{6} + \frac {6}{5} \, a^{2} c^{2} d^{5} x^{5} + 8 \, a^{3} c d^{3} e^{2} x^{5} + a^{4} d e^{4} x^{5} + 5 \, a^{3} c d^{4} e x^{4} + \frac {5}{2} \, a^{4} d^{2} e^{3} x^{4} + \frac {4}{3} \, a^{3} c d^{5} x^{3} + \frac {10}{3} \, a^{4} d^{3} e^{2} x^{3} + \frac {5}{2} \, a^{4} d^{4} e x^{2} + a^{4} d^{5} x \]
1/14*c^4*e^5*x^14 + 5/13*c^4*d*e^4*x^13 + 5/6*c^4*d^2*e^3*x^12 + 1/3*a*c^3 *e^5*x^12 + 10/11*c^4*d^3*e^2*x^11 + 20/11*a*c^3*d*e^4*x^11 + 1/2*c^4*d^4* e*x^10 + 4*a*c^3*d^2*e^3*x^10 + 3/5*a^2*c^2*e^5*x^10 + 1/9*c^4*d^5*x^9 + 4 0/9*a*c^3*d^3*e^2*x^9 + 10/3*a^2*c^2*d*e^4*x^9 + 5/2*a*c^3*d^4*e*x^8 + 15/ 2*a^2*c^2*d^2*e^3*x^8 + 1/2*a^3*c*e^5*x^8 + 4/7*a*c^3*d^5*x^7 + 60/7*a^2*c ^2*d^3*e^2*x^7 + 20/7*a^3*c*d*e^4*x^7 + 5*a^2*c^2*d^4*e*x^6 + 20/3*a^3*c*d ^2*e^3*x^6 + 1/6*a^4*e^5*x^6 + 6/5*a^2*c^2*d^5*x^5 + 8*a^3*c*d^3*e^2*x^5 + a^4*d*e^4*x^5 + 5*a^3*c*d^4*e*x^4 + 5/2*a^4*d^2*e^3*x^4 + 4/3*a^3*c*d^5*x ^3 + 10/3*a^4*d^3*e^2*x^3 + 5/2*a^4*d^4*e*x^2 + a^4*d^5*x
Time = 9.64 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.28 \[ \int (d+e x)^5 \left (a+c x^2\right )^4 \, dx=x^3\,\left (\frac {10\,a^4\,d^3\,e^2}{3}+\frac {4\,c\,a^3\,d^5}{3}\right )+x^{12}\,\left (\frac {5\,c^4\,d^2\,e^3}{6}+\frac {a\,c^3\,e^5}{3}\right )+x^5\,\left (a^4\,d\,e^4+8\,a^3\,c\,d^3\,e^2+\frac {6\,a^2\,c^2\,d^5}{5}\right )+x^6\,\left (\frac {a^4\,e^5}{6}+\frac {20\,a^3\,c\,d^2\,e^3}{3}+5\,a^2\,c^2\,d^4\,e\right )+x^9\,\left (\frac {10\,a^2\,c^2\,d\,e^4}{3}+\frac {40\,a\,c^3\,d^3\,e^2}{9}+\frac {c^4\,d^5}{9}\right )+x^{10}\,\left (\frac {3\,a^2\,c^2\,e^5}{5}+4\,a\,c^3\,d^2\,e^3+\frac {c^4\,d^4\,e}{2}\right )+a^4\,d^5\,x+\frac {c^4\,e^5\,x^{14}}{14}+\frac {5\,a^4\,d^4\,e\,x^2}{2}+\frac {5\,c^4\,d\,e^4\,x^{13}}{13}+\frac {4\,a\,c\,d\,x^7\,\left (5\,a^2\,e^4+15\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{7}+\frac {a\,c\,e\,x^8\,\left (a^2\,e^4+15\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{2}+\frac {5\,a^3\,d^2\,e\,x^4\,\left (2\,c\,d^2+a\,e^2\right )}{2}+\frac {10\,c^3\,d\,e^2\,x^{11}\,\left (c\,d^2+2\,a\,e^2\right )}{11} \]
x^3*((4*a^3*c*d^5)/3 + (10*a^4*d^3*e^2)/3) + x^12*((a*c^3*e^5)/3 + (5*c^4* d^2*e^3)/6) + x^5*(a^4*d*e^4 + (6*a^2*c^2*d^5)/5 + 8*a^3*c*d^3*e^2) + x^6* ((a^4*e^5)/6 + 5*a^2*c^2*d^4*e + (20*a^3*c*d^2*e^3)/3) + x^9*((c^4*d^5)/9 + (40*a*c^3*d^3*e^2)/9 + (10*a^2*c^2*d*e^4)/3) + x^10*((c^4*d^4*e)/2 + (3* a^2*c^2*e^5)/5 + 4*a*c^3*d^2*e^3) + a^4*d^5*x + (c^4*e^5*x^14)/14 + (5*a^4 *d^4*e*x^2)/2 + (5*c^4*d*e^4*x^13)/13 + (4*a*c*d*x^7*(5*a^2*e^4 + c^2*d^4 + 15*a*c*d^2*e^2))/7 + (a*c*e*x^8*(a^2*e^4 + 5*c^2*d^4 + 15*a*c*d^2*e^2))/ 2 + (5*a^3*d^2*e*x^4*(a*e^2 + 2*c*d^2))/2 + (10*c^3*d*e^2*x^11*(2*a*e^2 + c*d^2))/11