Integrand size = 17, antiderivative size = 278 \[ \int (d+e x)^7 \left (a+c x^2\right )^4 \, dx=\frac {\left (c d^2+a e^2\right )^4 (d+e x)^8}{8 e^9}-\frac {8 c d \left (c d^2+a e^2\right )^3 (d+e x)^9}{9 e^9}+\frac {2 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^{10}}{5 e^9}-\frac {8 c^2 d \left (c d^2+a e^2\right ) \left (7 c d^2+3 a e^2\right ) (d+e x)^{11}}{11 e^9}+\frac {c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^{12}}{6 e^9}-\frac {8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^{13}}{13 e^9}+\frac {2 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^{14}}{7 e^9}-\frac {8 c^4 d (d+e x)^{15}}{15 e^9}+\frac {c^4 (d+e x)^{16}}{16 e^9} \]
1/8*(a*e^2+c*d^2)^4*(e*x+d)^8/e^9-8/9*c*d*(a*e^2+c*d^2)^3*(e*x+d)^9/e^9+2/ 5*c*(a*e^2+c*d^2)^2*(a*e^2+7*c*d^2)*(e*x+d)^10/e^9-8/11*c^2*d*(a*e^2+c*d^2 )*(3*a*e^2+7*c*d^2)*(e*x+d)^11/e^9+1/6*c^2*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^ 2*d^4)*(e*x+d)^12/e^9-8/13*c^3*d*(3*a*e^2+7*c*d^2)*(e*x+d)^13/e^9+2/7*c^3* (a*e^2+7*c*d^2)*(e*x+d)^14/e^9-8/15*c^4*d*(e*x+d)^15/e^9+1/16*c^4*(e*x+d)^ 16/e^9
Time = 0.09 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.52 \[ \int (d+e x)^7 \left (a+c x^2\right )^4 \, dx=\frac {1}{8} a^4 x \left (8 d^7+28 d^6 e x+56 d^5 e^2 x^2+70 d^4 e^3 x^3+56 d^3 e^4 x^4+28 d^2 e^5 x^5+8 d e^6 x^6+e^7 x^7\right )+\frac {1}{90} a^3 c x^3 \left (120 d^7+630 d^6 e x+1512 d^5 e^2 x^2+2100 d^4 e^3 x^3+1800 d^3 e^4 x^4+945 d^2 e^5 x^5+280 d e^6 x^6+36 e^7 x^7\right )+\frac {1}{660} a^2 c^2 x^5 \left (792 d^7+4620 d^6 e x+11880 d^5 e^2 x^2+17325 d^4 e^3 x^3+15400 d^3 e^4 x^4+8316 d^2 e^5 x^5+2520 d e^6 x^6+330 e^7 x^7\right )+\frac {a c^3 x^7 \left (3432 d^7+21021 d^6 e x+56056 d^5 e^2 x^2+84084 d^4 e^3 x^3+76440 d^3 e^4 x^4+42042 d^2 e^5 x^5+12936 d e^6 x^6+1716 e^7 x^7\right )}{6006}+\frac {c^4 x^9 \left (11440 d^7+72072 d^6 e x+196560 d^5 e^2 x^2+300300 d^4 e^3 x^3+277200 d^3 e^4 x^4+154440 d^2 e^5 x^5+48048 d e^6 x^6+6435 e^7 x^7\right )}{102960} \]
(a^4*x*(8*d^7 + 28*d^6*e*x + 56*d^5*e^2*x^2 + 70*d^4*e^3*x^3 + 56*d^3*e^4* x^4 + 28*d^2*e^5*x^5 + 8*d*e^6*x^6 + e^7*x^7))/8 + (a^3*c*x^3*(120*d^7 + 6 30*d^6*e*x + 1512*d^5*e^2*x^2 + 2100*d^4*e^3*x^3 + 1800*d^3*e^4*x^4 + 945* d^2*e^5*x^5 + 280*d*e^6*x^6 + 36*e^7*x^7))/90 + (a^2*c^2*x^5*(792*d^7 + 46 20*d^6*e*x + 11880*d^5*e^2*x^2 + 17325*d^4*e^3*x^3 + 15400*d^3*e^4*x^4 + 8 316*d^2*e^5*x^5 + 2520*d*e^6*x^6 + 330*e^7*x^7))/660 + (a*c^3*x^7*(3432*d^ 7 + 21021*d^6*e*x + 56056*d^5*e^2*x^2 + 84084*d^4*e^3*x^3 + 76440*d^3*e^4* x^4 + 42042*d^2*e^5*x^5 + 12936*d*e^6*x^6 + 1716*e^7*x^7))/6006 + (c^4*x^9 *(11440*d^7 + 72072*d^6*e*x + 196560*d^5*e^2*x^2 + 300300*d^4*e^3*x^3 + 27 7200*d^3*e^4*x^4 + 154440*d^2*e^5*x^5 + 48048*d*e^6*x^6 + 6435*e^7*x^7))/1 02960
Time = 0.72 (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.
| \(\displaystyle \int \left (a+c x^2\right )^4 (d+e x)^7 \, dx\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \int \left (\frac {2 c^2 (d+e x)^{11} \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{e^8}+\frac {4 c^3 (d+e x)^{13} \left (a e^2+7 c d^2\right )}{e^8}-\frac {8 c^3 d (d+e x)^{12} \left (3 a e^2+7 c d^2\right )}{e^8}+\frac {8 c^2 d (d+e x)^{10} \left (-3 a e^2-7 c d^2\right ) \left (a e^2+c d^2\right )}{e^8}+\frac {4 c (d+e x)^9 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{e^8}-\frac {8 c d (d+e x)^8 \left (a e^2+c d^2\right )^3}{e^8}+\frac {(d+e x)^7 \left (a e^2+c d^2\right )^4}{e^8}+\frac {c^4 (d+e x)^{15}}{e^8}-\frac {8 c^4 d (d+e x)^{14}}{e^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^2 (d+e x)^{12} \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{6 e^9}+\frac {2 c^3 (d+e x)^{14} \left (a e^2+7 c d^2\right )}{7 e^9}-\frac {8 c^3 d (d+e x)^{13} \left (3 a e^2+7 c d^2\right )}{13 e^9}-\frac {8 c^2 d (d+e x)^{11} \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{11 e^9}+\frac {2 c (d+e x)^{10} \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{5 e^9}-\frac {8 c d (d+e x)^9 \left (a e^2+c d^2\right )^3}{9 e^9}+\frac {(d+e x)^8 \left (a e^2+c d^2\right )^4}{8 e^9}+\frac {c^4 (d+e x)^{16}}{16 e^9}-\frac {8 c^4 d (d+e x)^{15}}{15 e^9}\) |
((c*d^2 + a*e^2)^4*(d + e*x)^8)/(8*e^9) - (8*c*d*(c*d^2 + a*e^2)^3*(d + e* x)^9)/(9*e^9) + (2*c*(c*d^2 + a*e^2)^2*(7*c*d^2 + a*e^2)*(d + e*x)^10)/(5* e^9) - (8*c^2*d*(c*d^2 + a*e^2)*(7*c*d^2 + 3*a*e^2)*(d + e*x)^11)/(11*e^9) + (c^2*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)*(d + e*x)^12)/(6*e^9) - (8*c^3*d*(7*c*d^2 + 3*a*e^2)*(d + e*x)^13)/(13*e^9) + (2*c^3*(7*c*d^2 + a* e^2)*(d + e*x)^14)/(7*e^9) - (8*c^4*d*(d + e*x)^15)/(15*e^9) + (c^4*(d + e *x)^16)/(16*e^9)
3.5.88.3.1 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 = 2.14 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.80
| method | result | size |
| norman | \(a^{4} d^{7} x +\frac {7 d^{6} e \,a^{4} x^{2}}{2}+\left (7 d^{5} e^{2} a^{4}+\frac {4}{3} d^{7} c \,a^{3}\right ) x^{3}+\left (\frac {35}{4} d^{4} e^{3} a^{4}+7 d^{6} e c \,a^{3}\right ) x^{4}+\left (7 d^{3} e^{4} a^{4}+\frac {84}{5} d^{5} e^{2} c \,a^{3}+\frac {6}{5} d^{7} a^{2} c^{2}\right ) x^{5}+\left (\frac {7}{2} d^{2} e^{5} a^{4}+\frac {70}{3} d^{4} e^{3} c \,a^{3}+7 d^{6} e \,a^{2} c^{2}\right ) x^{6}+\left (d \,e^{6} a^{4}+20 d^{3} e^{4} c \,a^{3}+18 d^{5} e^{2} a^{2} c^{2}+\frac {4}{7} d^{7} c^{3} a \right ) x^{7}+\left (\frac {1}{8} e^{7} a^{4}+\frac {21}{2} d^{2} e^{5} c \,a^{3}+\frac {105}{4} d^{4} e^{3} a^{2} c^{2}+\frac {7}{2} d^{6} e \,c^{3} a \right ) x^{8}+\left (\frac {28}{9} d \,e^{6} c \,a^{3}+\frac {70}{3} d^{3} e^{4} a^{2} c^{2}+\frac {28}{3} d^{5} e^{2} c^{3} a +\frac {1}{9} d^{7} c^{4}\right ) x^{9}+\left (\frac {2}{5} e^{7} c \,a^{3}+\frac {63}{5} d^{2} e^{5} a^{2} c^{2}+14 d^{4} e^{3} c^{3} a +\frac {7}{10} d^{6} e \,c^{4}\right ) x^{10}+\left (\frac {42}{11} d \,e^{6} a^{2} c^{2}+\frac {140}{11} d^{3} e^{4} c^{3} a +\frac {21}{11} d^{5} e^{2} c^{4}\right ) x^{11}+\left (\frac {1}{2} e^{7} a^{2} c^{2}+7 d^{2} e^{5} c^{3} a +\frac {35}{12} d^{4} e^{3} c^{4}\right ) x^{12}+\left (\frac {28}{13} d \,e^{6} c^{3} a +\frac {35}{13} d^{3} e^{4} c^{4}\right ) x^{13}+\left (\frac {2}{7} e^{7} c^{3} a +\frac {3}{2} d^{2} e^{5} c^{4}\right ) x^{14}+\frac {7 d \,e^{6} c^{4} x^{15}}{15}+\frac {e^{7} c^{4} x^{16}}{16}\) | \(500\) |
| default | \(\frac {e^{7} c^{4} x^{16}}{16}+\frac {7 d \,e^{6} c^{4} x^{15}}{15}+\frac {\left (4 e^{7} c^{3} a +21 d^{2} e^{5} c^{4}\right ) x^{14}}{14}+\frac {\left (28 d \,e^{6} c^{3} a +35 d^{3} e^{4} c^{4}\right ) x^{13}}{13}+\frac {\left (6 e^{7} a^{2} c^{2}+84 d^{2} e^{5} c^{3} a +35 d^{4} e^{3} c^{4}\right ) x^{12}}{12}+\frac {\left (42 d \,e^{6} a^{2} c^{2}+140 d^{3} e^{4} c^{3} a +21 d^{5} e^{2} c^{4}\right ) x^{11}}{11}+\frac {\left (4 e^{7} c \,a^{3}+126 d^{2} e^{5} a^{2} c^{2}+140 d^{4} e^{3} c^{3} a +7 d^{6} e \,c^{4}\right ) x^{10}}{10}+\frac {\left (28 d \,e^{6} c \,a^{3}+210 d^{3} e^{4} a^{2} c^{2}+84 d^{5} e^{2} c^{3} a +d^{7} c^{4}\right ) x^{9}}{9}+\frac {\left (e^{7} a^{4}+84 d^{2} e^{5} c \,a^{3}+210 d^{4} e^{3} a^{2} c^{2}+28 d^{6} e \,c^{3} a \right ) x^{8}}{8}+\frac {\left (7 d \,e^{6} a^{4}+140 d^{3} e^{4} c \,a^{3}+126 d^{5} e^{2} a^{2} c^{2}+4 d^{7} c^{3} a \right ) x^{7}}{7}+\frac {\left (21 d^{2} e^{5} a^{4}+140 d^{4} e^{3} c \,a^{3}+42 d^{6} e \,a^{2} c^{2}\right ) x^{6}}{6}+\frac {\left (35 d^{3} e^{4} a^{4}+84 d^{5} e^{2} c \,a^{3}+6 d^{7} a^{2} c^{2}\right ) x^{5}}{5}+\frac {\left (35 d^{4} e^{3} a^{4}+28 d^{6} e c \,a^{3}\right ) x^{4}}{4}+\frac {\left (21 d^{5} e^{2} a^{4}+4 d^{7} c \,a^{3}\right ) x^{3}}{3}+\frac {7 d^{6} e \,a^{4} x^{2}}{2}+a^{4} d^{7} x\) | \(511\) |
| gosper | \(\frac {105}{4} x^{8} d^{4} e^{3} a^{2} c^{2}+\frac {7}{2} x^{8} d^{6} e \,c^{3} a +\frac {28}{9} x^{9} d \,e^{6} c \,a^{3}+14 x^{10} d^{4} e^{3} c^{3} a +\frac {42}{11} x^{11} d \,e^{6} a^{2} c^{2}+\frac {140}{11} x^{11} d^{3} e^{4} c^{3} a +7 x^{12} d^{2} e^{5} c^{3} a +\frac {28}{13} x^{13} d \,e^{6} c^{3} a +\frac {70}{3} x^{9} d^{3} e^{4} a^{2} c^{2}+7 x^{4} d^{6} e c \,a^{3}+\frac {84}{5} x^{5} d^{5} e^{2} c \,a^{3}+\frac {70}{3} x^{6} d^{4} e^{3} c \,a^{3}+7 x^{6} d^{6} e \,a^{2} c^{2}+20 x^{7} d^{3} e^{4} c \,a^{3}+\frac {21}{2} x^{8} d^{2} e^{5} c \,a^{3}+18 x^{7} d^{5} e^{2} a^{2} c^{2}+\frac {28}{3} x^{9} d^{5} e^{2} c^{3} a +\frac {63}{5} x^{10} d^{2} e^{5} a^{2} c^{2}+\frac {7}{2} d^{6} e \,a^{4} x^{2}+\frac {7}{15} d \,e^{6} c^{4} x^{15}+\frac {2}{5} x^{10} e^{7} c \,a^{3}+\frac {7}{10} x^{10} d^{6} e \,c^{4}+\frac {21}{11} x^{11} d^{5} e^{2} c^{4}+\frac {1}{2} x^{12} e^{7} a^{2} c^{2}+\frac {35}{12} x^{12} d^{4} e^{3} c^{4}+\frac {35}{13} x^{13} d^{3} e^{4} c^{4}+\frac {2}{7} x^{14} e^{7} c^{3} a +\frac {3}{2} x^{14} d^{2} e^{5} c^{4}+\frac {1}{9} x^{9} d^{7} c^{4}+a^{4} d^{7} x +\frac {1}{16} e^{7} c^{4} x^{16}+7 x^{3} d^{5} e^{2} a^{4}+\frac {4}{3} x^{3} d^{7} c \,a^{3}+\frac {1}{8} x^{8} e^{7} a^{4}+\frac {35}{4} x^{4} d^{4} e^{3} a^{4}+7 x^{5} d^{3} e^{4} a^{4}+\frac {6}{5} x^{5} d^{7} a^{2} c^{2}+\frac {7}{2} x^{6} d^{2} e^{5} a^{4}+x^{7} d \,e^{6} a^{4}+\frac {4}{7} x^{7} d^{7} c^{3} a\) | \(548\) |
| risch | \(\frac {105}{4} x^{8} d^{4} e^{3} a^{2} c^{2}+\frac {7}{2} x^{8} d^{6} e \,c^{3} a +\frac {28}{9} x^{9} d \,e^{6} c \,a^{3}+14 x^{10} d^{4} e^{3} c^{3} a +\frac {42}{11} x^{11} d \,e^{6} a^{2} c^{2}+\frac {140}{11} x^{11} d^{3} e^{4} c^{3} a +7 x^{12} d^{2} e^{5} c^{3} a +\frac {28}{13} x^{13} d \,e^{6} c^{3} a +\frac {70}{3} x^{9} d^{3} e^{4} a^{2} c^{2}+7 x^{4} d^{6} e c \,a^{3}+\frac {84}{5} x^{5} d^{5} e^{2} c \,a^{3}+\frac {70}{3} x^{6} d^{4} e^{3} c \,a^{3}+7 x^{6} d^{6} e \,a^{2} c^{2}+20 x^{7} d^{3} e^{4} c \,a^{3}+\frac {21}{2} x^{8} d^{2} e^{5} c \,a^{3}+18 x^{7} d^{5} e^{2} a^{2} c^{2}+\frac {28}{3} x^{9} d^{5} e^{2} c^{3} a +\frac {63}{5} x^{10} d^{2} e^{5} a^{2} c^{2}+\frac {7}{2} d^{6} e \,a^{4} x^{2}+\frac {7}{15} d \,e^{6} c^{4} x^{15}+\frac {2}{5} x^{10} e^{7} c \,a^{3}+\frac {7}{10} x^{10} d^{6} e \,c^{4}+\frac {21}{11} x^{11} d^{5} e^{2} c^{4}+\frac {1}{2} x^{12} e^{7} a^{2} c^{2}+\frac {35}{12} x^{12} d^{4} e^{3} c^{4}+\frac {35}{13} x^{13} d^{3} e^{4} c^{4}+\frac {2}{7} x^{14} e^{7} c^{3} a +\frac {3}{2} x^{14} d^{2} e^{5} c^{4}+\frac {1}{9} x^{9} d^{7} c^{4}+a^{4} d^{7} x +\frac {1}{16} e^{7} c^{4} x^{16}+7 x^{3} d^{5} e^{2} a^{4}+\frac {4}{3} x^{3} d^{7} c \,a^{3}+\frac {1}{8} x^{8} e^{7} a^{4}+\frac {35}{4} x^{4} d^{4} e^{3} a^{4}+7 x^{5} d^{3} e^{4} a^{4}+\frac {6}{5} x^{5} d^{7} a^{2} c^{2}+\frac {7}{2} x^{6} d^{2} e^{5} a^{4}+x^{7} d \,e^{6} a^{4}+\frac {4}{7} x^{7} d^{7} c^{3} a\) | \(548\) |
| parallelrisch | \(\frac {105}{4} x^{8} d^{4} e^{3} a^{2} c^{2}+\frac {7}{2} x^{8} d^{6} e \,c^{3} a +\frac {28}{9} x^{9} d \,e^{6} c \,a^{3}+14 x^{10} d^{4} e^{3} c^{3} a +\frac {42}{11} x^{11} d \,e^{6} a^{2} c^{2}+\frac {140}{11} x^{11} d^{3} e^{4} c^{3} a +7 x^{12} d^{2} e^{5} c^{3} a +\frac {28}{13} x^{13} d \,e^{6} c^{3} a +\frac {70}{3} x^{9} d^{3} e^{4} a^{2} c^{2}+7 x^{4} d^{6} e c \,a^{3}+\frac {84}{5} x^{5} d^{5} e^{2} c \,a^{3}+\frac {70}{3} x^{6} d^{4} e^{3} c \,a^{3}+7 x^{6} d^{6} e \,a^{2} c^{2}+20 x^{7} d^{3} e^{4} c \,a^{3}+\frac {21}{2} x^{8} d^{2} e^{5} c \,a^{3}+18 x^{7} d^{5} e^{2} a^{2} c^{2}+\frac {28}{3} x^{9} d^{5} e^{2} c^{3} a +\frac {63}{5} x^{10} d^{2} e^{5} a^{2} c^{2}+\frac {7}{2} d^{6} e \,a^{4} x^{2}+\frac {7}{15} d \,e^{6} c^{4} x^{15}+\frac {2}{5} x^{10} e^{7} c \,a^{3}+\frac {7}{10} x^{10} d^{6} e \,c^{4}+\frac {21}{11} x^{11} d^{5} e^{2} c^{4}+\frac {1}{2} x^{12} e^{7} a^{2} c^{2}+\frac {35}{12} x^{12} d^{4} e^{3} c^{4}+\frac {35}{13} x^{13} d^{3} e^{4} c^{4}+\frac {2}{7} x^{14} e^{7} c^{3} a +\frac {3}{2} x^{14} d^{2} e^{5} c^{4}+\frac {1}{9} x^{9} d^{7} c^{4}+a^{4} d^{7} x +\frac {1}{16} e^{7} c^{4} x^{16}+7 x^{3} d^{5} e^{2} a^{4}+\frac {4}{3} x^{3} d^{7} c \,a^{3}+\frac {1}{8} x^{8} e^{7} a^{4}+\frac {35}{4} x^{4} d^{4} e^{3} a^{4}+7 x^{5} d^{3} e^{4} a^{4}+\frac {6}{5} x^{5} d^{7} a^{2} c^{2}+\frac {7}{2} x^{6} d^{2} e^{5} a^{4}+x^{7} d \,e^{6} a^{4}+\frac {4}{7} x^{7} d^{7} c^{3} a\) | \(548\) |
a^4*d^7*x+7/2*d^6*e*a^4*x^2+(7*d^5*e^2*a^4+4/3*d^7*c*a^3)*x^3+(35/4*d^4*e^ 3*a^4+7*d^6*e*c*a^3)*x^4+(7*d^3*e^4*a^4+84/5*d^5*e^2*c*a^3+6/5*d^7*a^2*c^2 )*x^5+(7/2*d^2*e^5*a^4+70/3*d^4*e^3*c*a^3+7*d^6*e*a^2*c^2)*x^6+(d*e^6*a^4+ 20*d^3*e^4*c*a^3+18*d^5*e^2*a^2*c^2+4/7*d^7*c^3*a)*x^7+(1/8*e^7*a^4+21/2*d ^2*e^5*c*a^3+105/4*d^4*e^3*a^2*c^2+7/2*d^6*e*c^3*a)*x^8+(28/9*d*e^6*c*a^3+ 70/3*d^3*e^4*a^2*c^2+28/3*d^5*e^2*c^3*a+1/9*d^7*c^4)*x^9+(2/5*e^7*c*a^3+63 /5*d^2*e^5*a^2*c^2+14*d^4*e^3*c^3*a+7/10*d^6*e*c^4)*x^10+(42/11*d*e^6*a^2* c^2+140/11*d^3*e^4*c^3*a+21/11*d^5*e^2*c^4)*x^11+(1/2*e^7*a^2*c^2+7*d^2*e^ 5*c^3*a+35/12*d^4*e^3*c^4)*x^12+(28/13*d*e^6*c^3*a+35/13*d^3*e^4*c^4)*x^13 +(2/7*e^7*c^3*a+3/2*d^2*e^5*c^4)*x^14+7/15*d*e^6*c^4*x^15+1/16*e^7*c^4*x^1 6
Time = 0.35 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.83 \[ \int (d+e x)^7 \left (a+c x^2\right )^4 \, dx=\frac {1}{16} \, c^{4} e^{7} x^{16} + \frac {7}{15} \, c^{4} d e^{6} x^{15} + \frac {1}{14} \, {\left (21 \, c^{4} d^{2} e^{5} + 4 \, a c^{3} e^{7}\right )} x^{14} + \frac {7}{13} \, {\left (5 \, c^{4} d^{3} e^{4} + 4 \, a c^{3} d e^{6}\right )} x^{13} + \frac {7}{2} \, a^{4} d^{6} e x^{2} + \frac {1}{12} \, {\left (35 \, c^{4} d^{4} e^{3} + 84 \, a c^{3} d^{2} e^{5} + 6 \, a^{2} c^{2} e^{7}\right )} x^{12} + a^{4} d^{7} x + \frac {7}{11} \, {\left (3 \, c^{4} d^{5} e^{2} + 20 \, a c^{3} d^{3} e^{4} + 6 \, a^{2} c^{2} d e^{6}\right )} x^{11} + \frac {1}{10} \, {\left (7 \, c^{4} d^{6} e + 140 \, a c^{3} d^{4} e^{3} + 126 \, a^{2} c^{2} d^{2} e^{5} + 4 \, a^{3} c e^{7}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{7} + 84 \, a c^{3} d^{5} e^{2} + 210 \, a^{2} c^{2} d^{3} e^{4} + 28 \, a^{3} c d e^{6}\right )} x^{9} + \frac {1}{8} \, {\left (28 \, a c^{3} d^{6} e + 210 \, a^{2} c^{2} d^{4} e^{3} + 84 \, a^{3} c d^{2} e^{5} + a^{4} e^{7}\right )} x^{8} + \frac {1}{7} \, {\left (4 \, a c^{3} d^{7} + 126 \, a^{2} c^{2} d^{5} e^{2} + 140 \, a^{3} c d^{3} e^{4} + 7 \, a^{4} d e^{6}\right )} x^{7} + \frac {7}{6} \, {\left (6 \, a^{2} c^{2} d^{6} e + 20 \, a^{3} c d^{4} e^{3} + 3 \, a^{4} d^{2} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, a^{2} c^{2} d^{7} + 84 \, a^{3} c d^{5} e^{2} + 35 \, a^{4} d^{3} e^{4}\right )} x^{5} + \frac {7}{4} \, {\left (4 \, a^{3} c d^{6} e + 5 \, a^{4} d^{4} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (4 \, a^{3} c d^{7} + 21 \, a^{4} d^{5} e^{2}\right )} x^{3} \]
1/16*c^4*e^7*x^16 + 7/15*c^4*d*e^6*x^15 + 1/14*(21*c^4*d^2*e^5 + 4*a*c^3*e ^7)*x^14 + 7/13*(5*c^4*d^3*e^4 + 4*a*c^3*d*e^6)*x^13 + 7/2*a^4*d^6*e*x^2 + 1/12*(35*c^4*d^4*e^3 + 84*a*c^3*d^2*e^5 + 6*a^2*c^2*e^7)*x^12 + a^4*d^7*x + 7/11*(3*c^4*d^5*e^2 + 20*a*c^3*d^3*e^4 + 6*a^2*c^2*d*e^6)*x^11 + 1/10*( 7*c^4*d^6*e + 140*a*c^3*d^4*e^3 + 126*a^2*c^2*d^2*e^5 + 4*a^3*c*e^7)*x^10 + 1/9*(c^4*d^7 + 84*a*c^3*d^5*e^2 + 210*a^2*c^2*d^3*e^4 + 28*a^3*c*d*e^6)* x^9 + 1/8*(28*a*c^3*d^6*e + 210*a^2*c^2*d^4*e^3 + 84*a^3*c*d^2*e^5 + a^4*e ^7)*x^8 + 1/7*(4*a*c^3*d^7 + 126*a^2*c^2*d^5*e^2 + 140*a^3*c*d^3*e^4 + 7*a ^4*d*e^6)*x^7 + 7/6*(6*a^2*c^2*d^6*e + 20*a^3*c*d^4*e^3 + 3*a^4*d^2*e^5)*x ^6 + 1/5*(6*a^2*c^2*d^7 + 84*a^3*c*d^5*e^2 + 35*a^4*d^3*e^4)*x^5 + 7/4*(4* a^3*c*d^6*e + 5*a^4*d^4*e^3)*x^4 + 1/3*(4*a^3*c*d^7 + 21*a^4*d^5*e^2)*x^3
Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (272) = 544\).
Time = 0.06 (sec) , antiderivative size = 571, normalized size of antiderivative = 2.05 \[ \int (d+e x)^7 \left (a+c x^2\right )^4 \, dx=a^{4} d^{7} x + \frac {7 a^{4} d^{6} e x^{2}}{2} + \frac {7 c^{4} d e^{6} x^{15}}{15} + \frac {c^{4} e^{7} x^{16}}{16} + x^{14} \cdot \left (\frac {2 a c^{3} e^{7}}{7} + \frac {3 c^{4} d^{2} e^{5}}{2}\right ) + x^{13} \cdot \left (\frac {28 a c^{3} d e^{6}}{13} + \frac {35 c^{4} d^{3} e^{4}}{13}\right ) + x^{12} \left (\frac {a^{2} c^{2} e^{7}}{2} + 7 a c^{3} d^{2} e^{5} + \frac {35 c^{4} d^{4} e^{3}}{12}\right ) + x^{11} \cdot \left (\frac {42 a^{2} c^{2} d e^{6}}{11} + \frac {140 a c^{3} d^{3} e^{4}}{11} + \frac {21 c^{4} d^{5} e^{2}}{11}\right ) + x^{10} \cdot \left (\frac {2 a^{3} c e^{7}}{5} + \frac {63 a^{2} c^{2} d^{2} e^{5}}{5} + 14 a c^{3} d^{4} e^{3} + \frac {7 c^{4} d^{6} e}{10}\right ) + x^{9} \cdot \left (\frac {28 a^{3} c d e^{6}}{9} + \frac {70 a^{2} c^{2} d^{3} e^{4}}{3} + \frac {28 a c^{3} d^{5} e^{2}}{3} + \frac {c^{4} d^{7}}{9}\right ) + x^{8} \left (\frac {a^{4} e^{7}}{8} + \frac {21 a^{3} c d^{2} e^{5}}{2} + \frac {105 a^{2} c^{2} d^{4} e^{3}}{4} + \frac {7 a c^{3} d^{6} e}{2}\right ) + x^{7} \left (a^{4} d e^{6} + 20 a^{3} c d^{3} e^{4} + 18 a^{2} c^{2} d^{5} e^{2} + \frac {4 a c^{3} d^{7}}{7}\right ) + x^{6} \cdot \left (\frac {7 a^{4} d^{2} e^{5}}{2} + \frac {70 a^{3} c d^{4} e^{3}}{3} + 7 a^{2} c^{2} d^{6} e\right ) + x^{5} \cdot \left (7 a^{4} d^{3} e^{4} + \frac {84 a^{3} c d^{5} e^{2}}{5} + \frac {6 a^{2} c^{2} d^{7}}{5}\right ) + x^{4} \cdot \left (\frac {35 a^{4} d^{4} e^{3}}{4} + 7 a^{3} c d^{6} e\right ) + x^{3} \cdot \left (7 a^{4} d^{5} e^{2} + \frac {4 a^{3} c d^{7}}{3}\right ) \]
a**4*d**7*x + 7*a**4*d**6*e*x**2/2 + 7*c**4*d*e**6*x**15/15 + c**4*e**7*x* *16/16 + x**14*(2*a*c**3*e**7/7 + 3*c**4*d**2*e**5/2) + x**13*(28*a*c**3*d *e**6/13 + 35*c**4*d**3*e**4/13) + x**12*(a**2*c**2*e**7/2 + 7*a*c**3*d**2 *e**5 + 35*c**4*d**4*e**3/12) + x**11*(42*a**2*c**2*d*e**6/11 + 140*a*c**3 *d**3*e**4/11 + 21*c**4*d**5*e**2/11) + x**10*(2*a**3*c*e**7/5 + 63*a**2*c **2*d**2*e**5/5 + 14*a*c**3*d**4*e**3 + 7*c**4*d**6*e/10) + x**9*(28*a**3* c*d*e**6/9 + 70*a**2*c**2*d**3*e**4/3 + 28*a*c**3*d**5*e**2/3 + c**4*d**7/ 9) + x**8*(a**4*e**7/8 + 21*a**3*c*d**2*e**5/2 + 105*a**2*c**2*d**4*e**3/4 + 7*a*c**3*d**6*e/2) + x**7*(a**4*d*e**6 + 20*a**3*c*d**3*e**4 + 18*a**2* c**2*d**5*e**2 + 4*a*c**3*d**7/7) + x**6*(7*a**4*d**2*e**5/2 + 70*a**3*c*d **4*e**3/3 + 7*a**2*c**2*d**6*e) + x**5*(7*a**4*d**3*e**4 + 84*a**3*c*d**5 *e**2/5 + 6*a**2*c**2*d**7/5) + x**4*(35*a**4*d**4*e**3/4 + 7*a**3*c*d**6* e) + x**3*(7*a**4*d**5*e**2 + 4*a**3*c*d**7/3)
Time = 0.20 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.83 \[ \int (d+e x)^7 \left (a+c x^2\right )^4 \, dx=\frac {1}{16} \, c^{4} e^{7} x^{16} + \frac {7}{15} \, c^{4} d e^{6} x^{15} + \frac {1}{14} \, {\left (21 \, c^{4} d^{2} e^{5} + 4 \, a c^{3} e^{7}\right )} x^{14} + \frac {7}{13} \, {\left (5 \, c^{4} d^{3} e^{4} + 4 \, a c^{3} d e^{6}\right )} x^{13} + \frac {7}{2} \, a^{4} d^{6} e x^{2} + \frac {1}{12} \, {\left (35 \, c^{4} d^{4} e^{3} + 84 \, a c^{3} d^{2} e^{5} + 6 \, a^{2} c^{2} e^{7}\right )} x^{12} + a^{4} d^{7} x + \frac {7}{11} \, {\left (3 \, c^{4} d^{5} e^{2} + 20 \, a c^{3} d^{3} e^{4} + 6 \, a^{2} c^{2} d e^{6}\right )} x^{11} + \frac {1}{10} \, {\left (7 \, c^{4} d^{6} e + 140 \, a c^{3} d^{4} e^{3} + 126 \, a^{2} c^{2} d^{2} e^{5} + 4 \, a^{3} c e^{7}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{7} + 84 \, a c^{3} d^{5} e^{2} + 210 \, a^{2} c^{2} d^{3} e^{4} + 28 \, a^{3} c d e^{6}\right )} x^{9} + \frac {1}{8} \, {\left (28 \, a c^{3} d^{6} e + 210 \, a^{2} c^{2} d^{4} e^{3} + 84 \, a^{3} c d^{2} e^{5} + a^{4} e^{7}\right )} x^{8} + \frac {1}{7} \, {\left (4 \, a c^{3} d^{7} + 126 \, a^{2} c^{2} d^{5} e^{2} + 140 \, a^{3} c d^{3} e^{4} + 7 \, a^{4} d e^{6}\right )} x^{7} + \frac {7}{6} \, {\left (6 \, a^{2} c^{2} d^{6} e + 20 \, a^{3} c d^{4} e^{3} + 3 \, a^{4} d^{2} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, a^{2} c^{2} d^{7} + 84 \, a^{3} c d^{5} e^{2} + 35 \, a^{4} d^{3} e^{4}\right )} x^{5} + \frac {7}{4} \, {\left (4 \, a^{3} c d^{6} e + 5 \, a^{4} d^{4} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (4 \, a^{3} c d^{7} + 21 \, a^{4} d^{5} e^{2}\right )} x^{3} \]
1/16*c^4*e^7*x^16 + 7/15*c^4*d*e^6*x^15 + 1/14*(21*c^4*d^2*e^5 + 4*a*c^3*e ^7)*x^14 + 7/13*(5*c^4*d^3*e^4 + 4*a*c^3*d*e^6)*x^13 + 7/2*a^4*d^6*e*x^2 + 1/12*(35*c^4*d^4*e^3 + 84*a*c^3*d^2*e^5 + 6*a^2*c^2*e^7)*x^12 + a^4*d^7*x + 7/11*(3*c^4*d^5*e^2 + 20*a*c^3*d^3*e^4 + 6*a^2*c^2*d*e^6)*x^11 + 1/10*( 7*c^4*d^6*e + 140*a*c^3*d^4*e^3 + 126*a^2*c^2*d^2*e^5 + 4*a^3*c*e^7)*x^10 + 1/9*(c^4*d^7 + 84*a*c^3*d^5*e^2 + 210*a^2*c^2*d^3*e^4 + 28*a^3*c*d*e^6)* x^9 + 1/8*(28*a*c^3*d^6*e + 210*a^2*c^2*d^4*e^3 + 84*a^3*c*d^2*e^5 + a^4*e ^7)*x^8 + 1/7*(4*a*c^3*d^7 + 126*a^2*c^2*d^5*e^2 + 140*a^3*c*d^3*e^4 + 7*a ^4*d*e^6)*x^7 + 7/6*(6*a^2*c^2*d^6*e + 20*a^3*c*d^4*e^3 + 3*a^4*d^2*e^5)*x ^6 + 1/5*(6*a^2*c^2*d^7 + 84*a^3*c*d^5*e^2 + 35*a^4*d^3*e^4)*x^5 + 7/4*(4* a^3*c*d^6*e + 5*a^4*d^4*e^3)*x^4 + 1/3*(4*a^3*c*d^7 + 21*a^4*d^5*e^2)*x^3
Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (260) = 520\).
Time = 0.28 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.97 \[ \int (d+e x)^7 \left (a+c x^2\right )^4 \, dx=\frac {1}{16} \, c^{4} e^{7} x^{16} + \frac {7}{15} \, c^{4} d e^{6} x^{15} + \frac {3}{2} \, c^{4} d^{2} e^{5} x^{14} + \frac {2}{7} \, a c^{3} e^{7} x^{14} + \frac {35}{13} \, c^{4} d^{3} e^{4} x^{13} + \frac {28}{13} \, a c^{3} d e^{6} x^{13} + \frac {35}{12} \, c^{4} d^{4} e^{3} x^{12} + 7 \, a c^{3} d^{2} e^{5} x^{12} + \frac {1}{2} \, a^{2} c^{2} e^{7} x^{12} + \frac {21}{11} \, c^{4} d^{5} e^{2} x^{11} + \frac {140}{11} \, a c^{3} d^{3} e^{4} x^{11} + \frac {42}{11} \, a^{2} c^{2} d e^{6} x^{11} + \frac {7}{10} \, c^{4} d^{6} e x^{10} + 14 \, a c^{3} d^{4} e^{3} x^{10} + \frac {63}{5} \, a^{2} c^{2} d^{2} e^{5} x^{10} + \frac {2}{5} \, a^{3} c e^{7} x^{10} + \frac {1}{9} \, c^{4} d^{7} x^{9} + \frac {28}{3} \, a c^{3} d^{5} e^{2} x^{9} + \frac {70}{3} \, a^{2} c^{2} d^{3} e^{4} x^{9} + \frac {28}{9} \, a^{3} c d e^{6} x^{9} + \frac {7}{2} \, a c^{3} d^{6} e x^{8} + \frac {105}{4} \, a^{2} c^{2} d^{4} e^{3} x^{8} + \frac {21}{2} \, a^{3} c d^{2} e^{5} x^{8} + \frac {1}{8} \, a^{4} e^{7} x^{8} + \frac {4}{7} \, a c^{3} d^{7} x^{7} + 18 \, a^{2} c^{2} d^{5} e^{2} x^{7} + 20 \, a^{3} c d^{3} e^{4} x^{7} + a^{4} d e^{6} x^{7} + 7 \, a^{2} c^{2} d^{6} e x^{6} + \frac {70}{3} \, a^{3} c d^{4} e^{3} x^{6} + \frac {7}{2} \, a^{4} d^{2} e^{5} x^{6} + \frac {6}{5} \, a^{2} c^{2} d^{7} x^{5} + \frac {84}{5} \, a^{3} c d^{5} e^{2} x^{5} + 7 \, a^{4} d^{3} e^{4} x^{5} + 7 \, a^{3} c d^{6} e x^{4} + \frac {35}{4} \, a^{4} d^{4} e^{3} x^{4} + \frac {4}{3} \, a^{3} c d^{7} x^{3} + 7 \, a^{4} d^{5} e^{2} x^{3} + \frac {7}{2} \, a^{4} d^{6} e x^{2} + a^{4} d^{7} x \]
1/16*c^4*e^7*x^16 + 7/15*c^4*d*e^6*x^15 + 3/2*c^4*d^2*e^5*x^14 + 2/7*a*c^3 *e^7*x^14 + 35/13*c^4*d^3*e^4*x^13 + 28/13*a*c^3*d*e^6*x^13 + 35/12*c^4*d^ 4*e^3*x^12 + 7*a*c^3*d^2*e^5*x^12 + 1/2*a^2*c^2*e^7*x^12 + 21/11*c^4*d^5*e ^2*x^11 + 140/11*a*c^3*d^3*e^4*x^11 + 42/11*a^2*c^2*d*e^6*x^11 + 7/10*c^4* d^6*e*x^10 + 14*a*c^3*d^4*e^3*x^10 + 63/5*a^2*c^2*d^2*e^5*x^10 + 2/5*a^3*c *e^7*x^10 + 1/9*c^4*d^7*x^9 + 28/3*a*c^3*d^5*e^2*x^9 + 70/3*a^2*c^2*d^3*e^ 4*x^9 + 28/9*a^3*c*d*e^6*x^9 + 7/2*a*c^3*d^6*e*x^8 + 105/4*a^2*c^2*d^4*e^3 *x^8 + 21/2*a^3*c*d^2*e^5*x^8 + 1/8*a^4*e^7*x^8 + 4/7*a*c^3*d^7*x^7 + 18*a ^2*c^2*d^5*e^2*x^7 + 20*a^3*c*d^3*e^4*x^7 + a^4*d*e^6*x^7 + 7*a^2*c^2*d^6* e*x^6 + 70/3*a^3*c*d^4*e^3*x^6 + 7/2*a^4*d^2*e^5*x^6 + 6/5*a^2*c^2*d^7*x^5 + 84/5*a^3*c*d^5*e^2*x^5 + 7*a^4*d^3*e^4*x^5 + 7*a^3*c*d^6*e*x^4 + 35/4*a ^4*d^4*e^3*x^4 + 4/3*a^3*c*d^7*x^3 + 7*a^4*d^5*e^2*x^3 + 7/2*a^4*d^6*e*x^2 + a^4*d^7*x
Time = 9.57 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.78 \[ \int (d+e x)^7 \left (a+c x^2\right )^4 \, dx=x^3\,\left (7\,a^4\,d^5\,e^2+\frac {4\,c\,a^3\,d^7}{3}\right )+x^{14}\,\left (\frac {3\,c^4\,d^2\,e^5}{2}+\frac {2\,a\,c^3\,e^7}{7}\right )+x^7\,\left (a^4\,d\,e^6+20\,a^3\,c\,d^3\,e^4+18\,a^2\,c^2\,d^5\,e^2+\frac {4\,a\,c^3\,d^7}{7}\right )+x^8\,\left (\frac {a^4\,e^7}{8}+\frac {21\,a^3\,c\,d^2\,e^5}{2}+\frac {105\,a^2\,c^2\,d^4\,e^3}{4}+\frac {7\,a\,c^3\,d^6\,e}{2}\right )+x^9\,\left (\frac {28\,a^3\,c\,d\,e^6}{9}+\frac {70\,a^2\,c^2\,d^3\,e^4}{3}+\frac {28\,a\,c^3\,d^5\,e^2}{3}+\frac {c^4\,d^7}{9}\right )+x^{10}\,\left (\frac {2\,a^3\,c\,e^7}{5}+\frac {63\,a^2\,c^2\,d^2\,e^5}{5}+14\,a\,c^3\,d^4\,e^3+\frac {7\,c^4\,d^6\,e}{10}\right )+x^5\,\left (7\,a^4\,d^3\,e^4+\frac {84\,a^3\,c\,d^5\,e^2}{5}+\frac {6\,a^2\,c^2\,d^7}{5}\right )+x^{12}\,\left (\frac {a^2\,c^2\,e^7}{2}+7\,a\,c^3\,d^2\,e^5+\frac {35\,c^4\,d^4\,e^3}{12}\right )+a^4\,d^7\,x+\frac {c^4\,e^7\,x^{16}}{16}+\frac {7\,a^4\,d^6\,e\,x^2}{2}+\frac {7\,c^4\,d\,e^6\,x^{15}}{15}+\frac {7\,a^3\,d^4\,e\,x^4\,\left (4\,c\,d^2+5\,a\,e^2\right )}{4}+\frac {7\,c^3\,d\,e^4\,x^{13}\,\left (5\,c\,d^2+4\,a\,e^2\right )}{13}+\frac {7\,a^2\,d^2\,e\,x^6\,\left (3\,a^2\,e^4+20\,a\,c\,d^2\,e^2+6\,c^2\,d^4\right )}{6}+\frac {7\,c^2\,d\,e^2\,x^{11}\,\left (6\,a^2\,e^4+20\,a\,c\,d^2\,e^2+3\,c^2\,d^4\right )}{11} \]
x^3*((4*a^3*c*d^7)/3 + 7*a^4*d^5*e^2) + x^14*((2*a*c^3*e^7)/7 + (3*c^4*d^2 *e^5)/2) + x^7*((4*a*c^3*d^7)/7 + a^4*d*e^6 + 20*a^3*c*d^3*e^4 + 18*a^2*c^ 2*d^5*e^2) + x^8*((a^4*e^7)/8 + (21*a^3*c*d^2*e^5)/2 + (105*a^2*c^2*d^4*e^ 3)/4 + (7*a*c^3*d^6*e)/2) + x^9*((c^4*d^7)/9 + (28*a*c^3*d^5*e^2)/3 + (70* a^2*c^2*d^3*e^4)/3 + (28*a^3*c*d*e^6)/9) + x^10*((2*a^3*c*e^7)/5 + (7*c^4* d^6*e)/10 + 14*a*c^3*d^4*e^3 + (63*a^2*c^2*d^2*e^5)/5) + x^5*((6*a^2*c^2*d ^7)/5 + 7*a^4*d^3*e^4 + (84*a^3*c*d^5*e^2)/5) + x^12*((a^2*c^2*e^7)/2 + (3 5*c^4*d^4*e^3)/12 + 7*a*c^3*d^2*e^5) + a^4*d^7*x + (c^4*e^7*x^16)/16 + (7* a^4*d^6*e*x^2)/2 + (7*c^4*d*e^6*x^15)/15 + (7*a^3*d^4*e*x^4*(5*a*e^2 + 4*c *d^2))/4 + (7*c^3*d*e^4*x^13*(4*a*e^2 + 5*c*d^2))/13 + (7*a^2*d^2*e*x^6*(3 *a^2*e^4 + 6*c^2*d^4 + 20*a*c*d^2*e^2))/6 + (7*c^2*d*e^2*x^11*(6*a^2*e^4 + 3*c^2*d^4 + 20*a*c*d^2*e^2))/11