Integrand size = 17, antiderivative size = 161 \[ \int (d+e x)^3 \left (a+c x^2\right )^3 \, dx=a^3 d^3 x+a^2 d \left (c d^2+a e^2\right ) x^3+\frac {1}{4} a^3 e^3 x^4+\frac {3}{5} a c d \left (c d^2+3 a e^2\right ) x^5+\frac {1}{2} a^2 c e^3 x^6+\frac {1}{7} c^2 d \left (c d^2+9 a e^2\right ) x^7+\frac {3}{8} a c^2 e^3 x^8+\frac {1}{3} c^3 d e^2 x^9+\frac {1}{10} c^3 e^3 x^{10}+\frac {3 d^2 e \left (a+c x^2\right )^4}{8 c} \]
a^3*d^3*x+a^2*d*(a*e^2+c*d^2)*x^3+1/4*a^3*e^3*x^4+3/5*a*c*d*(3*a*e^2+c*d^2 )*x^5+1/2*a^2*c*e^3*x^6+1/7*c^2*d*(9*a*e^2+c*d^2)*x^7+3/8*a*c^2*e^3*x^8+1/ 3*c^3*d*e^2*x^9+1/10*c^3*e^3*x^10+3/8*d^2*e*(c*x^2+a)^4/c
Time = 0.05 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.96 \[ \int (d+e x)^3 \left (a+c x^2\right )^3 \, dx=\frac {1}{840} x \left (210 a^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+42 a^2 c x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+9 a c^2 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+c^3 x^6 \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )\right ) \]
(x*(210*a^3*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 42*a^2*c*x^2*(20 *d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 9*a*c^2*x^4*(56*d^3 + 140 *d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + c^3*x^6*(120*d^3 + 315*d^2*e*x + 280*d*e^2*x^2 + 84*e^3*x^3)))/840
Time = 0.34 (sec) , antiderivative size = 161, 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 = {475, 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.
| \(\displaystyle \int \left (a+c x^2\right )^3 (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 475 |
| \(\displaystyle \int \left (c^3 e^3 x^9+3 c^3 d e^2 x^8+3 a c^2 e^3 x^7+c^2 d \left (c d^2+9 a e^2\right ) x^6+3 a^2 c e^3 x^5+3 a c d \left (c d^2+3 a e^2\right ) x^4+a^3 e^3 x^3+3 a^2 d \left (c d^2+a e^2\right ) x^2+a^3 d^3\right )dx+\frac {3 d^2 e \left (a+c x^2\right )^4}{8 c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^3 d^3 x+\frac {1}{4} a^3 e^3 x^4+a^2 d x^3 \left (a e^2+c d^2\right )+\frac {1}{2} a^2 c e^3 x^6+\frac {1}{7} c^2 d x^7 \left (9 a e^2+c d^2\right )+\frac {3}{8} a c^2 e^3 x^8+\frac {3}{5} a c d x^5 \left (3 a e^2+c d^2\right )+\frac {3 d^2 e \left (a+c x^2\right )^4}{8 c}+\frac {1}{3} c^3 d e^2 x^9+\frac {1}{10} c^3 e^3 x^{10}\) |
a^3*d^3*x + a^2*d*(c*d^2 + a*e^2)*x^3 + (a^3*e^3*x^4)/4 + (3*a*c*d*(c*d^2 + 3*a*e^2)*x^5)/5 + (a^2*c*e^3*x^6)/2 + (c^2*d*(c*d^2 + 9*a*e^2)*x^7)/7 + (3*a*c^2*e^3*x^8)/8 + (c^3*d*e^2*x^9)/3 + (c^3*e^3*x^10)/10 + (3*d^2*e*(a + c*x^2)^4)/(8*c)
3.5.75.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp [d*n*c^(n - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Int[ExpandIntegran d[((c + d*x)^n - d*n*c^(n - 1)*x)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && IGtQ[n, 0] && LeQ[n, p]
Time = 2.17 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.14
| method | result | size |
| norman | \(\frac {c^{3} e^{3} x^{10}}{10}+\frac {c^{3} d \,e^{2} x^{9}}{3}+\left (\frac {3}{8} e^{3} c^{2} a +\frac {3}{8} d^{2} e \,c^{3}\right ) x^{8}+\left (\frac {9}{7} d \,e^{2} c^{2} a +\frac {1}{7} c^{3} d^{3}\right ) x^{7}+\left (\frac {1}{2} a^{2} c \,e^{3}+\frac {3}{2} c^{2} d^{2} a e \right ) x^{6}+\left (\frac {9}{5} d \,e^{2} a^{2} c +\frac {3}{5} d^{3} c^{2} a \right ) x^{5}+\left (\frac {1}{4} a^{3} e^{3}+\frac {9}{4} d^{2} e \,a^{2} c \right ) x^{4}+\left (d \,e^{2} a^{3}+a^{2} c \,d^{3}\right ) x^{3}+\frac {3 d^{2} e \,a^{3} x^{2}}{2}+a^{3} d^{3} x\) | \(183\) |
| gosper | \(\frac {1}{10} c^{3} e^{3} x^{10}+\frac {1}{3} c^{3} d \,e^{2} x^{9}+\frac {3}{8} a \,c^{2} e^{3} x^{8}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {9}{7} x^{7} d \,e^{2} c^{2} a +\frac {1}{7} x^{7} c^{3} d^{3}+\frac {1}{2} a^{2} c \,e^{3} x^{6}+\frac {3}{2} x^{6} c^{2} d^{2} a e +\frac {9}{5} x^{5} d \,e^{2} a^{2} c +\frac {3}{5} x^{5} d^{3} c^{2} a +\frac {1}{4} a^{3} e^{3} x^{4}+\frac {9}{4} x^{4} d^{2} e \,a^{2} c +a^{3} d \,e^{2} x^{3}+a^{2} c \,d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{3} x^{2}+a^{3} d^{3} x\) | \(189\) |
| default | \(\frac {c^{3} e^{3} x^{10}}{10}+\frac {c^{3} d \,e^{2} x^{9}}{3}+\frac {\left (3 e^{3} c^{2} a +3 d^{2} e \,c^{3}\right ) x^{8}}{8}+\frac {\left (9 d \,e^{2} c^{2} a +c^{3} d^{3}\right ) x^{7}}{7}+\frac {\left (3 a^{2} c \,e^{3}+9 c^{2} d^{2} a e \right ) x^{6}}{6}+\frac {\left (9 d \,e^{2} a^{2} c +3 d^{3} c^{2} a \right ) x^{5}}{5}+\frac {\left (a^{3} e^{3}+9 d^{2} e \,a^{2} c \right ) x^{4}}{4}+\frac {\left (3 d \,e^{2} a^{3}+3 a^{2} c \,d^{3}\right ) x^{3}}{3}+\frac {3 d^{2} e \,a^{3} x^{2}}{2}+a^{3} d^{3} x\) | \(189\) |
| risch | \(\frac {1}{10} c^{3} e^{3} x^{10}+\frac {1}{3} c^{3} d \,e^{2} x^{9}+\frac {3}{8} a \,c^{2} e^{3} x^{8}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {9}{7} x^{7} d \,e^{2} c^{2} a +\frac {1}{7} x^{7} c^{3} d^{3}+\frac {1}{2} a^{2} c \,e^{3} x^{6}+\frac {3}{2} x^{6} c^{2} d^{2} a e +\frac {9}{5} x^{5} d \,e^{2} a^{2} c +\frac {3}{5} x^{5} d^{3} c^{2} a +\frac {1}{4} a^{3} e^{3} x^{4}+\frac {9}{4} x^{4} d^{2} e \,a^{2} c +a^{3} d \,e^{2} x^{3}+a^{2} c \,d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{3} x^{2}+a^{3} d^{3} x\) | \(189\) |
| parallelrisch | \(\frac {1}{10} c^{3} e^{3} x^{10}+\frac {1}{3} c^{3} d \,e^{2} x^{9}+\frac {3}{8} a \,c^{2} e^{3} x^{8}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {9}{7} x^{7} d \,e^{2} c^{2} a +\frac {1}{7} x^{7} c^{3} d^{3}+\frac {1}{2} a^{2} c \,e^{3} x^{6}+\frac {3}{2} x^{6} c^{2} d^{2} a e +\frac {9}{5} x^{5} d \,e^{2} a^{2} c +\frac {3}{5} x^{5} d^{3} c^{2} a +\frac {1}{4} a^{3} e^{3} x^{4}+\frac {9}{4} x^{4} d^{2} e \,a^{2} c +a^{3} d \,e^{2} x^{3}+a^{2} c \,d^{3} x^{3}+\frac {3}{2} d^{2} e \,a^{3} x^{2}+a^{3} d^{3} x\) | \(189\) |
1/10*c^3*e^3*x^10+1/3*c^3*d*e^2*x^9+(3/8*e^3*c^2*a+3/8*d^2*e*c^3)*x^8+(9/7 *d*e^2*c^2*a+1/7*c^3*d^3)*x^7+(1/2*a^2*c*e^3+3/2*c^2*d^2*a*e)*x^6+(9/5*d*e ^2*a^2*c+3/5*d^3*c^2*a)*x^5+(1/4*a^3*e^3+9/4*d^2*e*a^2*c)*x^4+(a^3*d*e^2+a ^2*c*d^3)*x^3+3/2*d^2*e*a^3*x^2+a^3*d^3*x
Time = 0.42 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.12 \[ \int (d+e x)^3 \left (a+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, c^{3} d e^{2} x^{9} + \frac {3}{8} \, {\left (c^{3} d^{2} e + a c^{2} e^{3}\right )} x^{8} + \frac {3}{2} \, a^{3} d^{2} e x^{2} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, a c^{2} d e^{2}\right )} x^{7} + a^{3} d^{3} x + \frac {1}{2} \, {\left (3 \, a c^{2} d^{2} e + a^{2} c e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (9 \, a^{2} c d^{2} e + a^{3} e^{3}\right )} x^{4} + {\left (a^{2} c d^{3} + a^{3} d e^{2}\right )} x^{3} \]
1/10*c^3*e^3*x^10 + 1/3*c^3*d*e^2*x^9 + 3/8*(c^3*d^2*e + a*c^2*e^3)*x^8 + 3/2*a^3*d^2*e*x^2 + 1/7*(c^3*d^3 + 9*a*c^2*d*e^2)*x^7 + a^3*d^3*x + 1/2*(3 *a*c^2*d^2*e + a^2*c*e^3)*x^6 + 3/5*(a*c^2*d^3 + 3*a^2*c*d*e^2)*x^5 + 1/4* (9*a^2*c*d^2*e + a^3*e^3)*x^4 + (a^2*c*d^3 + a^3*d*e^2)*x^3
Time = 0.04 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.25 \[ \int (d+e x)^3 \left (a+c x^2\right )^3 \, dx=a^{3} d^{3} x + \frac {3 a^{3} d^{2} e x^{2}}{2} + \frac {c^{3} d e^{2} x^{9}}{3} + \frac {c^{3} e^{3} x^{10}}{10} + x^{8} \cdot \left (\frac {3 a c^{2} e^{3}}{8} + \frac {3 c^{3} d^{2} e}{8}\right ) + x^{7} \cdot \left (\frac {9 a c^{2} d e^{2}}{7} + \frac {c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac {a^{2} c e^{3}}{2} + \frac {3 a c^{2} d^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {9 a^{2} c d e^{2}}{5} + \frac {3 a c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{3} e^{3}}{4} + \frac {9 a^{2} c d^{2} e}{4}\right ) + x^{3} \left (a^{3} d e^{2} + a^{2} c d^{3}\right ) \]
a**3*d**3*x + 3*a**3*d**2*e*x**2/2 + c**3*d*e**2*x**9/3 + c**3*e**3*x**10/ 10 + x**8*(3*a*c**2*e**3/8 + 3*c**3*d**2*e/8) + x**7*(9*a*c**2*d*e**2/7 + c**3*d**3/7) + x**6*(a**2*c*e**3/2 + 3*a*c**2*d**2*e/2) + x**5*(9*a**2*c*d *e**2/5 + 3*a*c**2*d**3/5) + x**4*(a**3*e**3/4 + 9*a**2*c*d**2*e/4) + x**3 *(a**3*d*e**2 + a**2*c*d**3)
Time = 0.21 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.12 \[ \int (d+e x)^3 \left (a+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, c^{3} d e^{2} x^{9} + \frac {3}{8} \, {\left (c^{3} d^{2} e + a c^{2} e^{3}\right )} x^{8} + \frac {3}{2} \, a^{3} d^{2} e x^{2} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, a c^{2} d e^{2}\right )} x^{7} + a^{3} d^{3} x + \frac {1}{2} \, {\left (3 \, a c^{2} d^{2} e + a^{2} c e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (9 \, a^{2} c d^{2} e + a^{3} e^{3}\right )} x^{4} + {\left (a^{2} c d^{3} + a^{3} d e^{2}\right )} x^{3} \]
1/10*c^3*e^3*x^10 + 1/3*c^3*d*e^2*x^9 + 3/8*(c^3*d^2*e + a*c^2*e^3)*x^8 + 3/2*a^3*d^2*e*x^2 + 1/7*(c^3*d^3 + 9*a*c^2*d*e^2)*x^7 + a^3*d^3*x + 1/2*(3 *a*c^2*d^2*e + a^2*c*e^3)*x^6 + 3/5*(a*c^2*d^3 + 3*a^2*c*d*e^2)*x^5 + 1/4* (9*a^2*c*d^2*e + a^3*e^3)*x^4 + (a^2*c*d^3 + a^3*d*e^2)*x^3
Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.17 \[ \int (d+e x)^3 \left (a+c x^2\right )^3 \, dx=\frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, c^{3} d e^{2} x^{9} + \frac {3}{8} \, c^{3} d^{2} e x^{8} + \frac {3}{8} \, a c^{2} e^{3} x^{8} + \frac {1}{7} \, c^{3} d^{3} x^{7} + \frac {9}{7} \, a c^{2} d e^{2} x^{7} + \frac {3}{2} \, a c^{2} d^{2} e x^{6} + \frac {1}{2} \, a^{2} c e^{3} x^{6} + \frac {3}{5} \, a c^{2} d^{3} x^{5} + \frac {9}{5} \, a^{2} c d e^{2} x^{5} + \frac {9}{4} \, a^{2} c d^{2} e x^{4} + \frac {1}{4} \, a^{3} e^{3} x^{4} + a^{2} c d^{3} x^{3} + a^{3} d e^{2} x^{3} + \frac {3}{2} \, a^{3} d^{2} e x^{2} + a^{3} d^{3} x \]
1/10*c^3*e^3*x^10 + 1/3*c^3*d*e^2*x^9 + 3/8*c^3*d^2*e*x^8 + 3/8*a*c^2*e^3* x^8 + 1/7*c^3*d^3*x^7 + 9/7*a*c^2*d*e^2*x^7 + 3/2*a*c^2*d^2*e*x^6 + 1/2*a^ 2*c*e^3*x^6 + 3/5*a*c^2*d^3*x^5 + 9/5*a^2*c*d*e^2*x^5 + 9/4*a^2*c*d^2*e*x^ 4 + 1/4*a^3*e^3*x^4 + a^2*c*d^3*x^3 + a^3*d*e^2*x^3 + 3/2*a^3*d^2*e*x^2 + a^3*d^3*x
Time = 0.09 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.08 \[ \int (d+e x)^3 \left (a+c x^2\right )^3 \, dx=x^3\,\left (a^3\,d\,e^2+c\,a^2\,d^3\right )+x^4\,\left (\frac {a^3\,e^3}{4}+\frac {9\,c\,a^2\,d^2\,e}{4}\right )+x^7\,\left (\frac {c^3\,d^3}{7}+\frac {9\,a\,c^2\,d\,e^2}{7}\right )+x^8\,\left (\frac {3\,c^3\,d^2\,e}{8}+\frac {3\,a\,c^2\,e^3}{8}\right )+a^3\,d^3\,x+\frac {c^3\,e^3\,x^{10}}{10}+\frac {3\,a^3\,d^2\,e\,x^2}{2}+\frac {c^3\,d\,e^2\,x^9}{3}+\frac {3\,a\,c\,d\,x^5\,\left (c\,d^2+3\,a\,e^2\right )}{5}+\frac {a\,c\,e\,x^6\,\left (3\,c\,d^2+a\,e^2\right )}{2} \]
x^3*(a^2*c*d^3 + a^3*d*e^2) + x^4*((a^3*e^3)/4 + (9*a^2*c*d^2*e)/4) + x^7* ((c^3*d^3)/7 + (9*a*c^2*d*e^2)/7) + x^8*((3*a*c^2*e^3)/8 + (3*c^3*d^2*e)/8 ) + a^3*d^3*x + (c^3*e^3*x^10)/10 + (3*a^3*d^2*e*x^2)/2 + (c^3*d*e^2*x^9)/ 3 + (3*a*c*d*x^5*(3*a*e^2 + c*d^2))/5 + (a*c*e*x^6*(a*e^2 + 3*c*d^2))/2