Integrand size = 17, antiderivative size = 117 \[ \int (d+e x)^4 \left (a+c x^2\right )^2 \, dx=\frac {\left (c d^2+a e^2\right )^2 (d+e x)^5}{5 e^5}-\frac {2 c d \left (c d^2+a e^2\right ) (d+e x)^6}{3 e^5}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^7}{7 e^5}-\frac {c^2 d (d+e x)^8}{2 e^5}+\frac {c^2 (d+e x)^9}{9 e^5} \]
1/5*(a*e^2+c*d^2)^2*(e*x+d)^5/e^5-2/3*c*d*(a*e^2+c*d^2)*(e*x+d)^6/e^5+2/7* c*(a*e^2+3*c*d^2)*(e*x+d)^7/e^5-1/2*c^2*d*(e*x+d)^8/e^5+1/9*c^2*(e*x+d)^9/ e^5
Time = 0.02 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.43 \[ \int (d+e x)^4 \left (a+c x^2\right )^2 \, dx=a^2 d^4 x+2 a^2 d^3 e x^2+\frac {2}{3} a d^2 \left (c d^2+3 a e^2\right ) x^3+a d e \left (2 c d^2+a e^2\right ) x^4+\frac {1}{5} \left (c^2 d^4+12 a c d^2 e^2+a^2 e^4\right ) x^5+\frac {2}{3} c d e \left (c d^2+2 a e^2\right ) x^6+\frac {2}{7} c e^2 \left (3 c d^2+a e^2\right ) x^7+\frac {1}{2} c^2 d e^3 x^8+\frac {1}{9} c^2 e^4 x^9 \]
a^2*d^4*x + 2*a^2*d^3*e*x^2 + (2*a*d^2*(c*d^2 + 3*a*e^2)*x^3)/3 + a*d*e*(2 *c*d^2 + a*e^2)*x^4 + ((c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4)*x^5)/5 + (2*c* d*e*(c*d^2 + 2*a*e^2)*x^6)/3 + (2*c*e^2*(3*c*d^2 + a*e^2)*x^7)/7 + (c^2*d* e^3*x^8)/2 + (c^2*e^4*x^9)/9
Time = 0.33 (sec) , antiderivative size = 117, 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.
\(\displaystyle \int \left (a+c x^2\right )^2 (d+e x)^4 \, dx\) |
\(\Big \downarrow \) 476 |
\(\displaystyle \int \left (\frac {2 c (d+e x)^6 \left (a e^2+3 c d^2\right )}{e^4}-\frac {4 c d (d+e x)^5 \left (a e^2+c d^2\right )}{e^4}+\frac {(d+e x)^4 \left (a e^2+c d^2\right )^2}{e^4}+\frac {c^2 (d+e x)^8}{e^4}-\frac {4 c^2 d (d+e x)^7}{e^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 c (d+e x)^7 \left (a e^2+3 c d^2\right )}{7 e^5}-\frac {2 c d (d+e x)^6 \left (a e^2+c d^2\right )}{3 e^5}+\frac {(d+e x)^5 \left (a e^2+c d^2\right )^2}{5 e^5}+\frac {c^2 (d+e x)^9}{9 e^5}-\frac {c^2 d (d+e x)^8}{2 e^5}\) |
((c*d^2 + a*e^2)^2*(d + e*x)^5)/(5*e^5) - (2*c*d*(c*d^2 + a*e^2)*(d + e*x) ^6)/(3*e^5) + (2*c*(3*c*d^2 + a*e^2)*(d + e*x)^7)/(7*e^5) - (c^2*d*(d + e* x)^8)/(2*e^5) + (c^2*(d + e*x)^9)/(9*e^5)
3.5.60.3.1 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 = 165, normalized size of antiderivative = 1.41
method | result | size |
norman | \(\frac {e^{4} c^{2} x^{9}}{9}+\frac {d \,e^{3} c^{2} x^{8}}{2}+\left (\frac {2}{7} a c \,e^{4}+\frac {6}{7} d^{2} e^{2} c^{2}\right ) x^{7}+\left (\frac {4}{3} d \,e^{3} a c +\frac {2}{3} d^{3} e \,c^{2}\right ) x^{6}+\left (\frac {1}{5} a^{2} e^{4}+\frac {12}{5} a c \,d^{2} e^{2}+\frac {1}{5} c^{2} d^{4}\right ) x^{5}+\left (d \,e^{3} a^{2}+2 d^{3} e a c \right ) x^{4}+\left (2 d^{2} e^{2} a^{2}+\frac {2}{3} a c \,d^{4}\right ) x^{3}+2 a^{2} d^{3} e \,x^{2}+a^{2} d^{4} x\) | \(165\) |
default | \(\frac {e^{4} c^{2} x^{9}}{9}+\frac {d \,e^{3} c^{2} x^{8}}{2}+\frac {\left (2 a c \,e^{4}+6 d^{2} e^{2} c^{2}\right ) x^{7}}{7}+\frac {\left (8 d \,e^{3} a c +4 d^{3} e \,c^{2}\right ) x^{6}}{6}+\frac {\left (a^{2} e^{4}+12 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x^{5}}{5}+\frac {\left (4 d \,e^{3} a^{2}+8 d^{3} e a c \right ) x^{4}}{4}+\frac {\left (6 d^{2} e^{2} a^{2}+2 a c \,d^{4}\right ) x^{3}}{3}+2 a^{2} d^{3} e \,x^{2}+a^{2} d^{4} x\) | \(169\) |
gosper | \(\frac {1}{9} e^{4} c^{2} x^{9}+\frac {1}{2} d \,e^{3} c^{2} x^{8}+\frac {2}{7} x^{7} a c \,e^{4}+\frac {6}{7} x^{7} d^{2} e^{2} c^{2}+\frac {4}{3} x^{6} d \,e^{3} a c +\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {1}{5} x^{5} a^{2} e^{4}+\frac {12}{5} x^{5} a c \,d^{2} e^{2}+\frac {1}{5} c^{2} d^{4} x^{5}+a^{2} d \,e^{3} x^{4}+2 a c \,d^{3} e \,x^{4}+2 x^{3} d^{2} e^{2} a^{2}+\frac {2}{3} x^{3} a c \,d^{4}+2 a^{2} d^{3} e \,x^{2}+a^{2} d^{4} x\) | \(173\) |
risch | \(\frac {1}{9} e^{4} c^{2} x^{9}+\frac {1}{2} d \,e^{3} c^{2} x^{8}+\frac {2}{7} x^{7} a c \,e^{4}+\frac {6}{7} x^{7} d^{2} e^{2} c^{2}+\frac {4}{3} x^{6} d \,e^{3} a c +\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {1}{5} x^{5} a^{2} e^{4}+\frac {12}{5} x^{5} a c \,d^{2} e^{2}+\frac {1}{5} c^{2} d^{4} x^{5}+a^{2} d \,e^{3} x^{4}+2 a c \,d^{3} e \,x^{4}+2 x^{3} d^{2} e^{2} a^{2}+\frac {2}{3} x^{3} a c \,d^{4}+2 a^{2} d^{3} e \,x^{2}+a^{2} d^{4} x\) | \(173\) |
parallelrisch | \(\frac {1}{9} e^{4} c^{2} x^{9}+\frac {1}{2} d \,e^{3} c^{2} x^{8}+\frac {2}{7} x^{7} a c \,e^{4}+\frac {6}{7} x^{7} d^{2} e^{2} c^{2}+\frac {4}{3} x^{6} d \,e^{3} a c +\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {1}{5} x^{5} a^{2} e^{4}+\frac {12}{5} x^{5} a c \,d^{2} e^{2}+\frac {1}{5} c^{2} d^{4} x^{5}+a^{2} d \,e^{3} x^{4}+2 a c \,d^{3} e \,x^{4}+2 x^{3} d^{2} e^{2} a^{2}+\frac {2}{3} x^{3} a c \,d^{4}+2 a^{2} d^{3} e \,x^{2}+a^{2} d^{4} x\) | \(173\) |
1/9*e^4*c^2*x^9+1/2*d*e^3*c^2*x^8+(2/7*a*c*e^4+6/7*d^2*e^2*c^2)*x^7+(4/3*d *e^3*a*c+2/3*d^3*e*c^2)*x^6+(1/5*a^2*e^4+12/5*a*c*d^2*e^2+1/5*c^2*d^4)*x^5 +(a^2*d*e^3+2*a*c*d^3*e)*x^4+(2*d^2*e^2*a^2+2/3*a*c*d^4)*x^3+2*a^2*d^3*e*x ^2+a^2*d^4*x
Time = 0.26 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.39 \[ \int (d+e x)^4 \left (a+c x^2\right )^2 \, dx=\frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{2} \, c^{2} d e^{3} x^{8} + 2 \, a^{2} d^{3} e x^{2} + \frac {2}{7} \, {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{7} + a^{2} d^{4} x + \frac {2}{3} \, {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{4} + 12 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{5} + {\left (2 \, a c d^{3} e + a^{2} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (a c d^{4} + 3 \, a^{2} d^{2} e^{2}\right )} x^{3} \]
1/9*c^2*e^4*x^9 + 1/2*c^2*d*e^3*x^8 + 2*a^2*d^3*e*x^2 + 2/7*(3*c^2*d^2*e^2 + a*c*e^4)*x^7 + a^2*d^4*x + 2/3*(c^2*d^3*e + 2*a*c*d*e^3)*x^6 + 1/5*(c^2 *d^4 + 12*a*c*d^2*e^2 + a^2*e^4)*x^5 + (2*a*c*d^3*e + a^2*d*e^3)*x^4 + 2/3 *(a*c*d^4 + 3*a^2*d^2*e^2)*x^3
Time = 0.03 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.56 \[ \int (d+e x)^4 \left (a+c x^2\right )^2 \, dx=a^{2} d^{4} x + 2 a^{2} d^{3} e x^{2} + \frac {c^{2} d e^{3} x^{8}}{2} + \frac {c^{2} e^{4} x^{9}}{9} + x^{7} \cdot \left (\frac {2 a c e^{4}}{7} + \frac {6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \cdot \left (\frac {4 a c d e^{3}}{3} + \frac {2 c^{2} d^{3} e}{3}\right ) + x^{5} \left (\frac {a^{2} e^{4}}{5} + \frac {12 a c d^{2} e^{2}}{5} + \frac {c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 2 a c d^{3} e\right ) + x^{3} \cdot \left (2 a^{2} d^{2} e^{2} + \frac {2 a c d^{4}}{3}\right ) \]
a**2*d**4*x + 2*a**2*d**3*e*x**2 + c**2*d*e**3*x**8/2 + c**2*e**4*x**9/9 + x**7*(2*a*c*e**4/7 + 6*c**2*d**2*e**2/7) + x**6*(4*a*c*d*e**3/3 + 2*c**2* d**3*e/3) + x**5*(a**2*e**4/5 + 12*a*c*d**2*e**2/5 + c**2*d**4/5) + x**4*( a**2*d*e**3 + 2*a*c*d**3*e) + x**3*(2*a**2*d**2*e**2 + 2*a*c*d**4/3)
Time = 0.20 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.39 \[ \int (d+e x)^4 \left (a+c x^2\right )^2 \, dx=\frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{2} \, c^{2} d e^{3} x^{8} + 2 \, a^{2} d^{3} e x^{2} + \frac {2}{7} \, {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{7} + a^{2} d^{4} x + \frac {2}{3} \, {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{4} + 12 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{5} + {\left (2 \, a c d^{3} e + a^{2} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (a c d^{4} + 3 \, a^{2} d^{2} e^{2}\right )} x^{3} \]
1/9*c^2*e^4*x^9 + 1/2*c^2*d*e^3*x^8 + 2*a^2*d^3*e*x^2 + 2/7*(3*c^2*d^2*e^2 + a*c*e^4)*x^7 + a^2*d^4*x + 2/3*(c^2*d^3*e + 2*a*c*d*e^3)*x^6 + 1/5*(c^2 *d^4 + 12*a*c*d^2*e^2 + a^2*e^4)*x^5 + (2*a*c*d^3*e + a^2*d*e^3)*x^4 + 2/3 *(a*c*d^4 + 3*a^2*d^2*e^2)*x^3
Time = 0.27 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.47 \[ \int (d+e x)^4 \left (a+c x^2\right )^2 \, dx=\frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{2} \, c^{2} d e^{3} x^{8} + \frac {6}{7} \, c^{2} d^{2} e^{2} x^{7} + \frac {2}{7} \, a c e^{4} x^{7} + \frac {2}{3} \, c^{2} d^{3} e x^{6} + \frac {4}{3} \, a c d e^{3} x^{6} + \frac {1}{5} \, c^{2} d^{4} x^{5} + \frac {12}{5} \, a c d^{2} e^{2} x^{5} + \frac {1}{5} \, a^{2} e^{4} x^{5} + 2 \, a c d^{3} e x^{4} + a^{2} d e^{3} x^{4} + \frac {2}{3} \, a c d^{4} x^{3} + 2 \, a^{2} d^{2} e^{2} x^{3} + 2 \, a^{2} d^{3} e x^{2} + a^{2} d^{4} x \]
1/9*c^2*e^4*x^9 + 1/2*c^2*d*e^3*x^8 + 6/7*c^2*d^2*e^2*x^7 + 2/7*a*c*e^4*x^ 7 + 2/3*c^2*d^3*e*x^6 + 4/3*a*c*d*e^3*x^6 + 1/5*c^2*d^4*x^5 + 12/5*a*c*d^2 *e^2*x^5 + 1/5*a^2*e^4*x^5 + 2*a*c*d^3*e*x^4 + a^2*d*e^3*x^4 + 2/3*a*c*d^4 *x^3 + 2*a^2*d^2*e^2*x^3 + 2*a^2*d^3*e*x^2 + a^2*d^4*x
Time = 9.49 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.37 \[ \int (d+e x)^4 \left (a+c x^2\right )^2 \, dx=x^5\,\left (\frac {a^2\,e^4}{5}+\frac {12\,a\,c\,d^2\,e^2}{5}+\frac {c^2\,d^4}{5}\right )+x^3\,\left (2\,a^2\,d^2\,e^2+\frac {2\,c\,a\,d^4}{3}\right )+x^7\,\left (\frac {6\,c^2\,d^2\,e^2}{7}+\frac {2\,a\,c\,e^4}{7}\right )+a^2\,d^4\,x+\frac {c^2\,e^4\,x^9}{9}+2\,a^2\,d^3\,e\,x^2+\frac {c^2\,d\,e^3\,x^8}{2}+a\,d\,e\,x^4\,\left (2\,c\,d^2+a\,e^2\right )+\frac {2\,c\,d\,e\,x^6\,\left (c\,d^2+2\,a\,e^2\right )}{3} \]