Integrand size = 17, antiderivative size = 190 \[ \int (d+e x)^5 \left (a+c x^2\right )^3 \, dx=\frac {\left (c d^2+a e^2\right )^3 (d+e x)^6}{6 e^7}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^7}{7 e^7}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^8}{8 e^7}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^9}{9 e^7}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{10}}{10 e^7}-\frac {6 c^3 d (d+e x)^{11}}{11 e^7}+\frac {c^3 (d+e x)^{12}}{12 e^7} \]
1/6*(a*e^2+c*d^2)^3*(e*x+d)^6/e^7-6/7*c*d*(a*e^2+c*d^2)^2*(e*x+d)^7/e^7+3/ 8*c*(a*e^2+c*d^2)*(a*e^2+5*c*d^2)*(e*x+d)^8/e^7-4/9*c^2*d*(3*a*e^2+5*c*d^2 )*(e*x+d)^9/e^7+3/10*c^2*(a*e^2+5*c*d^2)*(e*x+d)^10/e^7-6/11*c^3*d*(e*x+d) ^11/e^7+1/12*c^3*(e*x+d)^12/e^7
Time = 0.06 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.33 \[ \int (d+e x)^5 \left (a+c x^2\right )^3 \, dx=\frac {1}{6} a^3 x \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+\frac {1}{420} a c^2 x^5 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+\frac {c^3 x^7 \left (792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right )}{5544}+a^2 c \left (d^5 x^3+\frac {15}{4} d^4 e x^4+6 d^3 e^2 x^5+5 d^2 e^3 x^6+\frac {15}{7} d e^4 x^7+\frac {3 e^5 x^8}{8}\right ) \]
(a^3*x*(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))/6 + (a*c^2*x^5*(252*d^5 + 1050*d^4*e*x + 1800*d^3*e^2*x^2 + 1 575*d^2*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5))/420 + (c^3*x^7*(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))/5544 + a^2*c*(d^5*x^3 + (15*d^4*e*x^4)/4 + 6*d^3*e^2*x^5 + 5*d^2 *e^3*x^6 + (15*d*e^4*x^7)/7 + (3*e^5*x^8)/8)
Time = 0.47 (sec) , antiderivative size = 190, 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.
| \(\displaystyle \int \left (a+c x^2\right )^3 (d+e x)^5 \, dx\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \int \left (\frac {3 c^2 (d+e x)^9 \left (a e^2+5 c d^2\right )}{e^6}-\frac {4 c^2 d (d+e x)^8 \left (3 a e^2+5 c d^2\right )}{e^6}+\frac {3 c (d+e x)^7 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^6}-\frac {6 c d (d+e x)^6 \left (a e^2+c d^2\right )^2}{e^6}+\frac {(d+e x)^5 \left (a e^2+c d^2\right )^3}{e^6}+\frac {c^3 (d+e x)^{11}}{e^6}-\frac {6 c^3 d (d+e x)^{10}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 c^2 (d+e x)^{10} \left (a e^2+5 c d^2\right )}{10 e^7}-\frac {4 c^2 d (d+e x)^9 \left (3 a e^2+5 c d^2\right )}{9 e^7}+\frac {3 c (d+e x)^8 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{8 e^7}-\frac {6 c d (d+e x)^7 \left (a e^2+c d^2\right )^2}{7 e^7}+\frac {(d+e x)^6 \left (a e^2+c d^2\right )^3}{6 e^7}+\frac {c^3 (d+e x)^{12}}{12 e^7}-\frac {6 c^3 d (d+e x)^{11}}{11 e^7}\) |
((c*d^2 + a*e^2)^3*(d + e*x)^6)/(6*e^7) - (6*c*d*(c*d^2 + a*e^2)^2*(d + e* x)^7)/(7*e^7) + (3*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^8)/(8*e^7 ) - (4*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^9)/(9*e^7) + (3*c^2*(5*c*d^2 + a*e^2)*(d + e*x)^10)/(10*e^7) - (6*c^3*d*(d + e*x)^11)/(11*e^7) + (c^3*(d + e*x)^12)/(12*e^7)
3.5.73.3.1 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 = 2.33 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.49
| method | result | size |
| norman | \(\frac {e^{5} c^{3} x^{12}}{12}+\frac {5 d \,e^{4} c^{3} x^{11}}{11}+\left (\frac {3}{10} e^{5} c^{2} a +d^{2} e^{3} c^{3}\right ) x^{10}+\left (\frac {5}{3} d \,e^{4} c^{2} a +\frac {10}{9} d^{3} e^{2} c^{3}\right ) x^{9}+\left (\frac {3}{8} e^{5} a^{2} c +\frac {15}{4} d^{2} e^{3} c^{2} a +\frac {5}{8} d^{4} e \,c^{3}\right ) x^{8}+\left (\frac {15}{7} d \,e^{4} a^{2} c +\frac {30}{7} d^{3} e^{2} c^{2} a +\frac {1}{7} d^{5} c^{3}\right ) x^{7}+\left (\frac {1}{6} a^{3} e^{5}+5 d^{2} e^{3} a^{2} c +\frac {5}{2} d^{4} e \,c^{2} a \right ) x^{6}+\left (d \,e^{4} a^{3}+6 d^{3} e^{2} a^{2} c +\frac {3}{5} d^{5} c^{2} a \right ) x^{5}+\left (\frac {5}{2} d^{2} e^{3} a^{3}+\frac {15}{4} d^{4} e \,a^{2} c \right ) x^{4}+\left (\frac {10}{3} d^{3} e^{2} a^{3}+d^{5} a^{2} c \right ) x^{3}+\frac {5 d^{4} e \,a^{3} x^{2}}{2}+a^{3} d^{5} x\) | \(284\) |
| default | \(\frac {e^{5} c^{3} x^{12}}{12}+\frac {5 d \,e^{4} c^{3} x^{11}}{11}+\frac {\left (3 e^{5} c^{2} a +10 d^{2} e^{3} c^{3}\right ) x^{10}}{10}+\frac {\left (15 d \,e^{4} c^{2} a +10 d^{3} e^{2} c^{3}\right ) x^{9}}{9}+\frac {\left (3 e^{5} a^{2} c +30 d^{2} e^{3} c^{2} a +5 d^{4} e \,c^{3}\right ) x^{8}}{8}+\frac {\left (15 d \,e^{4} a^{2} c +30 d^{3} e^{2} c^{2} a +d^{5} c^{3}\right ) x^{7}}{7}+\frac {\left (a^{3} e^{5}+30 d^{2} e^{3} a^{2} c +15 d^{4} e \,c^{2} a \right ) x^{6}}{6}+\frac {\left (5 d \,e^{4} a^{3}+30 d^{3} e^{2} a^{2} c +3 d^{5} c^{2} a \right ) x^{5}}{5}+\frac {\left (10 d^{2} e^{3} a^{3}+15 d^{4} e \,a^{2} c \right ) x^{4}}{4}+\frac {\left (10 d^{3} e^{2} a^{3}+3 d^{5} a^{2} c \right ) x^{3}}{3}+\frac {5 d^{4} e \,a^{3} x^{2}}{2}+a^{3} d^{5} x\) | \(293\) |
| gosper | \(\frac {1}{12} e^{5} c^{3} x^{12}+\frac {5}{11} d \,e^{4} c^{3} x^{11}+\frac {3}{10} x^{10} e^{5} c^{2} a +x^{10} d^{2} e^{3} c^{3}+\frac {5}{3} x^{9} d \,e^{4} c^{2} a +\frac {10}{9} x^{9} d^{3} e^{2} c^{3}+\frac {3}{8} x^{8} e^{5} a^{2} c +\frac {15}{4} x^{8} d^{2} e^{3} c^{2} a +\frac {5}{8} x^{8} d^{4} e \,c^{3}+\frac {15}{7} x^{7} d \,e^{4} a^{2} c +\frac {30}{7} x^{7} d^{3} e^{2} c^{2} a +\frac {1}{7} x^{7} d^{5} c^{3}+\frac {1}{6} x^{6} a^{3} e^{5}+5 x^{6} d^{2} e^{3} a^{2} c +\frac {5}{2} x^{6} d^{4} e \,c^{2} a +x^{5} d \,e^{4} a^{3}+6 x^{5} d^{3} e^{2} a^{2} c +\frac {3}{5} x^{5} d^{5} c^{2} a +\frac {5}{2} x^{4} d^{2} e^{3} a^{3}+\frac {15}{4} x^{4} d^{4} e \,a^{2} c +\frac {10}{3} x^{3} d^{3} e^{2} a^{3}+x^{3} d^{5} a^{2} c +\frac {5}{2} d^{4} e \,a^{3} x^{2}+a^{3} d^{5} x\) | \(304\) |
| risch | \(\frac {1}{12} e^{5} c^{3} x^{12}+\frac {5}{11} d \,e^{4} c^{3} x^{11}+\frac {3}{10} x^{10} e^{5} c^{2} a +x^{10} d^{2} e^{3} c^{3}+\frac {5}{3} x^{9} d \,e^{4} c^{2} a +\frac {10}{9} x^{9} d^{3} e^{2} c^{3}+\frac {3}{8} x^{8} e^{5} a^{2} c +\frac {15}{4} x^{8} d^{2} e^{3} c^{2} a +\frac {5}{8} x^{8} d^{4} e \,c^{3}+\frac {15}{7} x^{7} d \,e^{4} a^{2} c +\frac {30}{7} x^{7} d^{3} e^{2} c^{2} a +\frac {1}{7} x^{7} d^{5} c^{3}+\frac {1}{6} x^{6} a^{3} e^{5}+5 x^{6} d^{2} e^{3} a^{2} c +\frac {5}{2} x^{6} d^{4} e \,c^{2} a +x^{5} d \,e^{4} a^{3}+6 x^{5} d^{3} e^{2} a^{2} c +\frac {3}{5} x^{5} d^{5} c^{2} a +\frac {5}{2} x^{4} d^{2} e^{3} a^{3}+\frac {15}{4} x^{4} d^{4} e \,a^{2} c +\frac {10}{3} x^{3} d^{3} e^{2} a^{3}+x^{3} d^{5} a^{2} c +\frac {5}{2} d^{4} e \,a^{3} x^{2}+a^{3} d^{5} x\) | \(304\) |
| parallelrisch | \(\frac {1}{12} e^{5} c^{3} x^{12}+\frac {5}{11} d \,e^{4} c^{3} x^{11}+\frac {3}{10} x^{10} e^{5} c^{2} a +x^{10} d^{2} e^{3} c^{3}+\frac {5}{3} x^{9} d \,e^{4} c^{2} a +\frac {10}{9} x^{9} d^{3} e^{2} c^{3}+\frac {3}{8} x^{8} e^{5} a^{2} c +\frac {15}{4} x^{8} d^{2} e^{3} c^{2} a +\frac {5}{8} x^{8} d^{4} e \,c^{3}+\frac {15}{7} x^{7} d \,e^{4} a^{2} c +\frac {30}{7} x^{7} d^{3} e^{2} c^{2} a +\frac {1}{7} x^{7} d^{5} c^{3}+\frac {1}{6} x^{6} a^{3} e^{5}+5 x^{6} d^{2} e^{3} a^{2} c +\frac {5}{2} x^{6} d^{4} e \,c^{2} a +x^{5} d \,e^{4} a^{3}+6 x^{5} d^{3} e^{2} a^{2} c +\frac {3}{5} x^{5} d^{5} c^{2} a +\frac {5}{2} x^{4} d^{2} e^{3} a^{3}+\frac {15}{4} x^{4} d^{4} e \,a^{2} c +\frac {10}{3} x^{3} d^{3} e^{2} a^{3}+x^{3} d^{5} a^{2} c +\frac {5}{2} d^{4} e \,a^{3} x^{2}+a^{3} d^{5} x\) | \(304\) |
1/12*e^5*c^3*x^12+5/11*d*e^4*c^3*x^11+(3/10*e^5*c^2*a+d^2*e^3*c^3)*x^10+(5 /3*d*e^4*c^2*a+10/9*d^3*e^2*c^3)*x^9+(3/8*e^5*a^2*c+15/4*d^2*e^3*c^2*a+5/8 *d^4*e*c^3)*x^8+(15/7*d*e^4*a^2*c+30/7*d^3*e^2*c^2*a+1/7*d^5*c^3)*x^7+(1/6 *a^3*e^5+5*d^2*e^3*a^2*c+5/2*d^4*e*c^2*a)*x^6+(d*e^4*a^3+6*d^3*e^2*a^2*c+3 /5*d^5*c^2*a)*x^5+(5/2*d^2*e^3*a^3+15/4*d^4*e*a^2*c)*x^4+(10/3*d^3*e^2*a^3 +d^5*a^2*c)*x^3+5/2*d^4*e*a^3*x^2+a^3*d^5*x
Time = 0.26 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.54 \[ \int (d+e x)^5 \left (a+c x^2\right )^3 \, dx=\frac {1}{12} \, c^{3} e^{5} x^{12} + \frac {5}{11} \, c^{3} d e^{4} x^{11} + \frac {1}{10} \, {\left (10 \, c^{3} d^{2} e^{3} + 3 \, a c^{2} e^{5}\right )} x^{10} + \frac {5}{2} \, a^{3} d^{4} e x^{2} + \frac {5}{9} \, {\left (2 \, c^{3} d^{3} e^{2} + 3 \, a c^{2} d e^{4}\right )} x^{9} + a^{3} d^{5} x + \frac {1}{8} \, {\left (5 \, c^{3} d^{4} e + 30 \, a c^{2} d^{2} e^{3} + 3 \, a^{2} c e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{5} + 30 \, a c^{2} d^{3} e^{2} + 15 \, a^{2} c d e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (15 \, a c^{2} d^{4} e + 30 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, a c^{2} d^{5} + 30 \, a^{2} c d^{3} e^{2} + 5 \, a^{3} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (3 \, a^{2} c d^{4} e + 2 \, a^{3} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a^{2} c d^{5} + 10 \, a^{3} d^{3} e^{2}\right )} x^{3} \]
1/12*c^3*e^5*x^12 + 5/11*c^3*d*e^4*x^11 + 1/10*(10*c^3*d^2*e^3 + 3*a*c^2*e ^5)*x^10 + 5/2*a^3*d^4*e*x^2 + 5/9*(2*c^3*d^3*e^2 + 3*a*c^2*d*e^4)*x^9 + a ^3*d^5*x + 1/8*(5*c^3*d^4*e + 30*a*c^2*d^2*e^3 + 3*a^2*c*e^5)*x^8 + 1/7*(c ^3*d^5 + 30*a*c^2*d^3*e^2 + 15*a^2*c*d*e^4)*x^7 + 1/6*(15*a*c^2*d^4*e + 30 *a^2*c*d^2*e^3 + a^3*e^5)*x^6 + 1/5*(3*a*c^2*d^5 + 30*a^2*c*d^3*e^2 + 5*a^ 3*d*e^4)*x^5 + 5/4*(3*a^2*c*d^4*e + 2*a^3*d^2*e^3)*x^4 + 1/3*(3*a^2*c*d^5 + 10*a^3*d^3*e^2)*x^3
Time = 0.04 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.69 \[ \int (d+e x)^5 \left (a+c x^2\right )^3 \, dx=a^{3} d^{5} x + \frac {5 a^{3} d^{4} e x^{2}}{2} + \frac {5 c^{3} d e^{4} x^{11}}{11} + \frac {c^{3} e^{5} x^{12}}{12} + x^{10} \cdot \left (\frac {3 a c^{2} e^{5}}{10} + c^{3} d^{2} e^{3}\right ) + x^{9} \cdot \left (\frac {5 a c^{2} d e^{4}}{3} + \frac {10 c^{3} d^{3} e^{2}}{9}\right ) + x^{8} \cdot \left (\frac {3 a^{2} c e^{5}}{8} + \frac {15 a c^{2} d^{2} e^{3}}{4} + \frac {5 c^{3} d^{4} e}{8}\right ) + x^{7} \cdot \left (\frac {15 a^{2} c d e^{4}}{7} + \frac {30 a c^{2} d^{3} e^{2}}{7} + \frac {c^{3} d^{5}}{7}\right ) + x^{6} \left (\frac {a^{3} e^{5}}{6} + 5 a^{2} c d^{2} e^{3} + \frac {5 a c^{2} d^{4} e}{2}\right ) + x^{5} \left (a^{3} d e^{4} + 6 a^{2} c d^{3} e^{2} + \frac {3 a c^{2} d^{5}}{5}\right ) + x^{4} \cdot \left (\frac {5 a^{3} d^{2} e^{3}}{2} + \frac {15 a^{2} c d^{4} e}{4}\right ) + x^{3} \cdot \left (\frac {10 a^{3} d^{3} e^{2}}{3} + a^{2} c d^{5}\right ) \]
a**3*d**5*x + 5*a**3*d**4*e*x**2/2 + 5*c**3*d*e**4*x**11/11 + c**3*e**5*x* *12/12 + x**10*(3*a*c**2*e**5/10 + c**3*d**2*e**3) + x**9*(5*a*c**2*d*e**4 /3 + 10*c**3*d**3*e**2/9) + x**8*(3*a**2*c*e**5/8 + 15*a*c**2*d**2*e**3/4 + 5*c**3*d**4*e/8) + x**7*(15*a**2*c*d*e**4/7 + 30*a*c**2*d**3*e**2/7 + c* *3*d**5/7) + x**6*(a**3*e**5/6 + 5*a**2*c*d**2*e**3 + 5*a*c**2*d**4*e/2) + x**5*(a**3*d*e**4 + 6*a**2*c*d**3*e**2 + 3*a*c**2*d**5/5) + x**4*(5*a**3* d**2*e**3/2 + 15*a**2*c*d**4*e/4) + x**3*(10*a**3*d**3*e**2/3 + a**2*c*d** 5)
Time = 0.20 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.54 \[ \int (d+e x)^5 \left (a+c x^2\right )^3 \, dx=\frac {1}{12} \, c^{3} e^{5} x^{12} + \frac {5}{11} \, c^{3} d e^{4} x^{11} + \frac {1}{10} \, {\left (10 \, c^{3} d^{2} e^{3} + 3 \, a c^{2} e^{5}\right )} x^{10} + \frac {5}{2} \, a^{3} d^{4} e x^{2} + \frac {5}{9} \, {\left (2 \, c^{3} d^{3} e^{2} + 3 \, a c^{2} d e^{4}\right )} x^{9} + a^{3} d^{5} x + \frac {1}{8} \, {\left (5 \, c^{3} d^{4} e + 30 \, a c^{2} d^{2} e^{3} + 3 \, a^{2} c e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{5} + 30 \, a c^{2} d^{3} e^{2} + 15 \, a^{2} c d e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (15 \, a c^{2} d^{4} e + 30 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, a c^{2} d^{5} + 30 \, a^{2} c d^{3} e^{2} + 5 \, a^{3} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (3 \, a^{2} c d^{4} e + 2 \, a^{3} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a^{2} c d^{5} + 10 \, a^{3} d^{3} e^{2}\right )} x^{3} \]
1/12*c^3*e^5*x^12 + 5/11*c^3*d*e^4*x^11 + 1/10*(10*c^3*d^2*e^3 + 3*a*c^2*e ^5)*x^10 + 5/2*a^3*d^4*e*x^2 + 5/9*(2*c^3*d^3*e^2 + 3*a*c^2*d*e^4)*x^9 + a ^3*d^5*x + 1/8*(5*c^3*d^4*e + 30*a*c^2*d^2*e^3 + 3*a^2*c*e^5)*x^8 + 1/7*(c ^3*d^5 + 30*a*c^2*d^3*e^2 + 15*a^2*c*d*e^4)*x^7 + 1/6*(15*a*c^2*d^4*e + 30 *a^2*c*d^2*e^3 + a^3*e^5)*x^6 + 1/5*(3*a*c^2*d^5 + 30*a^2*c*d^3*e^2 + 5*a^ 3*d*e^4)*x^5 + 5/4*(3*a^2*c*d^4*e + 2*a^3*d^2*e^3)*x^4 + 1/3*(3*a^2*c*d^5 + 10*a^3*d^3*e^2)*x^3
Time = 0.27 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.59 \[ \int (d+e x)^5 \left (a+c x^2\right )^3 \, dx=\frac {1}{12} \, c^{3} e^{5} x^{12} + \frac {5}{11} \, c^{3} d e^{4} x^{11} + c^{3} d^{2} e^{3} x^{10} + \frac {3}{10} \, a c^{2} e^{5} x^{10} + \frac {10}{9} \, c^{3} d^{3} e^{2} x^{9} + \frac {5}{3} \, a c^{2} d e^{4} x^{9} + \frac {5}{8} \, c^{3} d^{4} e x^{8} + \frac {15}{4} \, a c^{2} d^{2} e^{3} x^{8} + \frac {3}{8} \, a^{2} c e^{5} x^{8} + \frac {1}{7} \, c^{3} d^{5} x^{7} + \frac {30}{7} \, a c^{2} d^{3} e^{2} x^{7} + \frac {15}{7} \, a^{2} c d e^{4} x^{7} + \frac {5}{2} \, a c^{2} d^{4} e x^{6} + 5 \, a^{2} c d^{2} e^{3} x^{6} + \frac {1}{6} \, a^{3} e^{5} x^{6} + \frac {3}{5} \, a c^{2} d^{5} x^{5} + 6 \, a^{2} c d^{3} e^{2} x^{5} + a^{3} d e^{4} x^{5} + \frac {15}{4} \, a^{2} c d^{4} e x^{4} + \frac {5}{2} \, a^{3} d^{2} e^{3} x^{4} + a^{2} c d^{5} x^{3} + \frac {10}{3} \, a^{3} d^{3} e^{2} x^{3} + \frac {5}{2} \, a^{3} d^{4} e x^{2} + a^{3} d^{5} x \]
1/12*c^3*e^5*x^12 + 5/11*c^3*d*e^4*x^11 + c^3*d^2*e^3*x^10 + 3/10*a*c^2*e^ 5*x^10 + 10/9*c^3*d^3*e^2*x^9 + 5/3*a*c^2*d*e^4*x^9 + 5/8*c^3*d^4*e*x^8 + 15/4*a*c^2*d^2*e^3*x^8 + 3/8*a^2*c*e^5*x^8 + 1/7*c^3*d^5*x^7 + 30/7*a*c^2* d^3*e^2*x^7 + 15/7*a^2*c*d*e^4*x^7 + 5/2*a*c^2*d^4*e*x^6 + 5*a^2*c*d^2*e^3 *x^6 + 1/6*a^3*e^5*x^6 + 3/5*a*c^2*d^5*x^5 + 6*a^2*c*d^3*e^2*x^5 + a^3*d*e ^4*x^5 + 15/4*a^2*c*d^4*e*x^4 + 5/2*a^3*d^2*e^3*x^4 + a^2*c*d^5*x^3 + 10/3 *a^3*d^3*e^2*x^3 + 5/2*a^3*d^4*e*x^2 + a^3*d^5*x
Time = 9.48 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.48 \[ \int (d+e x)^5 \left (a+c x^2\right )^3 \, dx=x^5\,\left (a^3\,d\,e^4+6\,a^2\,c\,d^3\,e^2+\frac {3\,a\,c^2\,d^5}{5}\right )+x^6\,\left (\frac {a^3\,e^5}{6}+5\,a^2\,c\,d^2\,e^3+\frac {5\,a\,c^2\,d^4\,e}{2}\right )+x^7\,\left (\frac {15\,a^2\,c\,d\,e^4}{7}+\frac {30\,a\,c^2\,d^3\,e^2}{7}+\frac {c^3\,d^5}{7}\right )+x^8\,\left (\frac {3\,a^2\,c\,e^5}{8}+\frac {15\,a\,c^2\,d^2\,e^3}{4}+\frac {5\,c^3\,d^4\,e}{8}\right )+x^3\,\left (\frac {10\,a^3\,d^3\,e^2}{3}+c\,a^2\,d^5\right )+x^{10}\,\left (c^3\,d^2\,e^3+\frac {3\,a\,c^2\,e^5}{10}\right )+a^3\,d^5\,x+\frac {c^3\,e^5\,x^{12}}{12}+\frac {5\,a^3\,d^4\,e\,x^2}{2}+\frac {5\,c^3\,d\,e^4\,x^{11}}{11}+\frac {5\,a^2\,d^2\,e\,x^4\,\left (3\,c\,d^2+2\,a\,e^2\right )}{4}+\frac {5\,c^2\,d\,e^2\,x^9\,\left (2\,c\,d^2+3\,a\,e^2\right )}{9} \]
x^5*((3*a*c^2*d^5)/5 + a^3*d*e^4 + 6*a^2*c*d^3*e^2) + x^6*((a^3*e^5)/6 + 5 *a^2*c*d^2*e^3 + (5*a*c^2*d^4*e)/2) + x^7*((c^3*d^5)/7 + (30*a*c^2*d^3*e^2 )/7 + (15*a^2*c*d*e^4)/7) + x^8*((3*a^2*c*e^5)/8 + (5*c^3*d^4*e)/8 + (15*a *c^2*d^2*e^3)/4) + x^3*(a^2*c*d^5 + (10*a^3*d^3*e^2)/3) + x^10*((3*a*c^2*e ^5)/10 + c^3*d^2*e^3) + a^3*d^5*x + (c^3*e^5*x^12)/12 + (5*a^3*d^4*e*x^2)/ 2 + (5*c^3*d*e^4*x^11)/11 + (5*a^2*d^2*e*x^4*(2*a*e^2 + 3*c*d^2))/4 + (5*c ^2*d*e^2*x^9*(3*a*e^2 + 2*c*d^2))/9