Integrand size = 17, antiderivative size = 188 \[ \int (d+e x)^4 \left (a+c x^2\right )^3 \, dx=\frac {\left (c d^2+a e^2\right )^3 (d+e x)^5}{5 e^7}-\frac {c d \left (c d^2+a e^2\right )^2 (d+e x)^6}{e^7}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^7}{7 e^7}-\frac {c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^8}{2 e^7}+\frac {c^2 \left (5 c d^2+a e^2\right ) (d+e x)^9}{3 e^7}-\frac {3 c^3 d (d+e x)^{10}}{5 e^7}+\frac {c^3 (d+e x)^{11}}{11 e^7} \]
1/5*(a*e^2+c*d^2)^3*(e*x+d)^5/e^7-c*d*(a*e^2+c*d^2)^2*(e*x+d)^6/e^7+3/7*c* (a*e^2+c*d^2)*(a*e^2+5*c*d^2)*(e*x+d)^7/e^7-1/2*c^2*d*(3*a*e^2+5*c*d^2)*(e *x+d)^8/e^7+1/3*c^2*(a*e^2+5*c*d^2)*(e*x+d)^9/e^7-3/5*c^3*d*(e*x+d)^10/e^7 +1/11*c^3*(e*x+d)^11/e^7
Time = 0.05 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.10 \[ \int (d+e x)^4 \left (a+c x^2\right )^3 \, dx=\frac {1}{210} a c^2 x^5 \left (126 d^4+420 d^3 e x+540 d^2 e^2 x^2+315 d e^3 x^3+70 e^4 x^4\right )+\frac {c^3 x^7 \left (330 d^4+1155 d^3 e x+1540 d^2 e^2 x^2+924 d e^3 x^3+210 e^4 x^4\right )}{2310}+a^3 \left (d^4 x+2 d^3 e x^2+2 d^2 e^2 x^3+d e^3 x^4+\frac {e^4 x^5}{5}\right )+a^2 c \left (d^4 x^3+3 d^3 e x^4+\frac {18}{5} d^2 e^2 x^5+2 d e^3 x^6+\frac {3 e^4 x^7}{7}\right ) \]
(a*c^2*x^5*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e ^4*x^4))/210 + (c^3*x^7*(330*d^4 + 1155*d^3*e*x + 1540*d^2*e^2*x^2 + 924*d *e^3*x^3 + 210*e^4*x^4))/2310 + a^3*(d^4*x + 2*d^3*e*x^2 + 2*d^2*e^2*x^3 + d*e^3*x^4 + (e^4*x^5)/5) + a^2*c*(d^4*x^3 + 3*d^3*e*x^4 + (18*d^2*e^2*x^5 )/5 + 2*d*e^3*x^6 + (3*e^4*x^7)/7)
Time = 0.44 (sec) , antiderivative size = 188, 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.
| \(\displaystyle \int \left (a+c x^2\right )^3 (d+e x)^4 \, dx\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \int \left (\frac {3 c^2 (d+e x)^8 \left (a e^2+5 c d^2\right )}{e^6}-\frac {4 c^2 d (d+e x)^7 \left (3 a e^2+5 c d^2\right )}{e^6}+\frac {3 c (d+e x)^6 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^6}-\frac {6 c d (d+e x)^5 \left (a e^2+c d^2\right )^2}{e^6}+\frac {(d+e x)^4 \left (a e^2+c d^2\right )^3}{e^6}+\frac {c^3 (d+e x)^{10}}{e^6}-\frac {6 c^3 d (d+e x)^9}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^2 (d+e x)^9 \left (a e^2+5 c d^2\right )}{3 e^7}-\frac {c^2 d (d+e x)^8 \left (3 a e^2+5 c d^2\right )}{2 e^7}+\frac {3 c (d+e x)^7 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7}-\frac {c d (d+e x)^6 \left (a e^2+c d^2\right )^2}{e^7}+\frac {(d+e x)^5 \left (a e^2+c d^2\right )^3}{5 e^7}+\frac {c^3 (d+e x)^{11}}{11 e^7}-\frac {3 c^3 d (d+e x)^{10}}{5 e^7}\) |
((c*d^2 + a*e^2)^3*(d + e*x)^5)/(5*e^7) - (c*d*(c*d^2 + a*e^2)^2*(d + e*x) ^6)/e^7 + (3*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^7)/(7*e^7) - (c ^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^8)/(2*e^7) + (c^2*(5*c*d^2 + a*e^2)*(d + e*x)^9)/(3*e^7) - (3*c^3*d*(d + e*x)^10)/(5*e^7) + (c^3*(d + e*x)^11)/(1 1*e^7)
3.5.74.3.1 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 = 2.32 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.24
| method | result | size |
| norman | \(\frac {c^{3} e^{4} x^{11}}{11}+\frac {2 d \,e^{3} c^{3} x^{10}}{5}+\left (\frac {1}{3} e^{4} c^{2} a +\frac {2}{3} d^{2} e^{2} c^{3}\right ) x^{9}+\left (\frac {3}{2} d \,e^{3} c^{2} a +\frac {1}{2} d^{3} e \,c^{3}\right ) x^{8}+\left (\frac {3}{7} e^{4} a^{2} c +\frac {18}{7} d^{2} e^{2} c^{2} a +\frac {1}{7} d^{4} c^{3}\right ) x^{7}+\left (2 d \,e^{3} a^{2} c +2 d^{3} e \,c^{2} a \right ) x^{6}+\left (\frac {1}{5} e^{4} a^{3}+\frac {18}{5} d^{2} e^{2} a^{2} c +\frac {3}{5} d^{4} c^{2} a \right ) x^{5}+\left (d \,e^{3} a^{3}+3 d^{3} e \,a^{2} c \right ) x^{4}+\left (2 d^{2} e^{2} a^{3}+a^{2} c \,d^{4}\right ) x^{3}+2 d^{3} e \,a^{3} x^{2}+d^{4} x \,a^{3}\) | \(234\) |
| default | \(\frac {c^{3} e^{4} x^{11}}{11}+\frac {2 d \,e^{3} c^{3} x^{10}}{5}+\frac {\left (3 e^{4} c^{2} a +6 d^{2} e^{2} c^{3}\right ) x^{9}}{9}+\frac {\left (12 d \,e^{3} c^{2} a +4 d^{3} e \,c^{3}\right ) x^{8}}{8}+\frac {\left (3 e^{4} a^{2} c +18 d^{2} e^{2} c^{2} a +d^{4} c^{3}\right ) x^{7}}{7}+\frac {\left (12 d \,e^{3} a^{2} c +12 d^{3} e \,c^{2} a \right ) x^{6}}{6}+\frac {\left (e^{4} a^{3}+18 d^{2} e^{2} a^{2} c +3 d^{4} c^{2} a \right ) x^{5}}{5}+\frac {\left (4 d \,e^{3} a^{3}+12 d^{3} e \,a^{2} c \right ) x^{4}}{4}+\frac {\left (6 d^{2} e^{2} a^{3}+3 a^{2} c \,d^{4}\right ) x^{3}}{3}+2 d^{3} e \,a^{3} x^{2}+d^{4} x \,a^{3}\) | \(241\) |
| gosper | \(\frac {1}{11} c^{3} e^{4} x^{11}+\frac {2}{5} d \,e^{3} c^{3} x^{10}+\frac {1}{3} x^{9} e^{4} c^{2} a +\frac {2}{3} x^{9} d^{2} e^{2} c^{3}+\frac {3}{2} x^{8} d \,e^{3} c^{2} a +\frac {1}{2} x^{8} d^{3} e \,c^{3}+\frac {3}{7} x^{7} e^{4} a^{2} c +\frac {18}{7} x^{7} d^{2} e^{2} c^{2} a +\frac {1}{7} x^{7} d^{4} c^{3}+2 a^{2} c d \,e^{3} x^{6}+2 a \,c^{2} d^{3} e \,x^{6}+\frac {1}{5} x^{5} e^{4} a^{3}+\frac {18}{5} x^{5} d^{2} e^{2} a^{2} c +\frac {3}{5} x^{5} d^{4} c^{2} a +a^{3} d \,e^{3} x^{4}+3 a^{2} c \,d^{3} e \,x^{4}+2 a^{3} d^{2} e^{2} x^{3}+a^{2} c \,d^{4} x^{3}+2 d^{3} e \,a^{3} x^{2}+d^{4} x \,a^{3}\) | \(247\) |
| risch | \(\frac {1}{11} c^{3} e^{4} x^{11}+\frac {2}{5} d \,e^{3} c^{3} x^{10}+\frac {1}{3} x^{9} e^{4} c^{2} a +\frac {2}{3} x^{9} d^{2} e^{2} c^{3}+\frac {3}{2} x^{8} d \,e^{3} c^{2} a +\frac {1}{2} x^{8} d^{3} e \,c^{3}+\frac {3}{7} x^{7} e^{4} a^{2} c +\frac {18}{7} x^{7} d^{2} e^{2} c^{2} a +\frac {1}{7} x^{7} d^{4} c^{3}+2 a^{2} c d \,e^{3} x^{6}+2 a \,c^{2} d^{3} e \,x^{6}+\frac {1}{5} x^{5} e^{4} a^{3}+\frac {18}{5} x^{5} d^{2} e^{2} a^{2} c +\frac {3}{5} x^{5} d^{4} c^{2} a +a^{3} d \,e^{3} x^{4}+3 a^{2} c \,d^{3} e \,x^{4}+2 a^{3} d^{2} e^{2} x^{3}+a^{2} c \,d^{4} x^{3}+2 d^{3} e \,a^{3} x^{2}+d^{4} x \,a^{3}\) | \(247\) |
| parallelrisch | \(\frac {1}{11} c^{3} e^{4} x^{11}+\frac {2}{5} d \,e^{3} c^{3} x^{10}+\frac {1}{3} x^{9} e^{4} c^{2} a +\frac {2}{3} x^{9} d^{2} e^{2} c^{3}+\frac {3}{2} x^{8} d \,e^{3} c^{2} a +\frac {1}{2} x^{8} d^{3} e \,c^{3}+\frac {3}{7} x^{7} e^{4} a^{2} c +\frac {18}{7} x^{7} d^{2} e^{2} c^{2} a +\frac {1}{7} x^{7} d^{4} c^{3}+2 a^{2} c d \,e^{3} x^{6}+2 a \,c^{2} d^{3} e \,x^{6}+\frac {1}{5} x^{5} e^{4} a^{3}+\frac {18}{5} x^{5} d^{2} e^{2} a^{2} c +\frac {3}{5} x^{5} d^{4} c^{2} a +a^{3} d \,e^{3} x^{4}+3 a^{2} c \,d^{3} e \,x^{4}+2 a^{3} d^{2} e^{2} x^{3}+a^{2} c \,d^{4} x^{3}+2 d^{3} e \,a^{3} x^{2}+d^{4} x \,a^{3}\) | \(247\) |
1/11*c^3*e^4*x^11+2/5*d*e^3*c^3*x^10+(1/3*e^4*c^2*a+2/3*d^2*e^2*c^3)*x^9+( 3/2*d*e^3*c^2*a+1/2*d^3*e*c^3)*x^8+(3/7*e^4*a^2*c+18/7*d^2*e^2*c^2*a+1/7*d ^4*c^3)*x^7+(2*a^2*c*d*e^3+2*a*c^2*d^3*e)*x^6+(1/5*e^4*a^3+18/5*d^2*e^2*a^ 2*c+3/5*d^4*c^2*a)*x^5+(a^3*d*e^3+3*a^2*c*d^3*e)*x^4+(2*a^3*d^2*e^2+a^2*c* d^4)*x^3+2*d^3*e*a^3*x^2+d^4*x*a^3
Time = 0.28 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.23 \[ \int (d+e x)^4 \left (a+c x^2\right )^3 \, dx=\frac {1}{11} \, c^{3} e^{4} x^{11} + \frac {2}{5} \, c^{3} d e^{3} x^{10} + \frac {1}{3} \, {\left (2 \, c^{3} d^{2} e^{2} + a c^{2} e^{4}\right )} x^{9} + 2 \, a^{3} d^{3} e x^{2} + \frac {1}{2} \, {\left (c^{3} d^{3} e + 3 \, a c^{2} d e^{3}\right )} x^{8} + a^{3} d^{4} x + \frac {1}{7} \, {\left (c^{3} d^{4} + 18 \, a c^{2} d^{2} e^{2} + 3 \, a^{2} c e^{4}\right )} x^{7} + 2 \, {\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, a c^{2} d^{4} + 18 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} x^{5} + {\left (3 \, a^{2} c d^{3} e + a^{3} d e^{3}\right )} x^{4} + {\left (a^{2} c d^{4} + 2 \, a^{3} d^{2} e^{2}\right )} x^{3} \]
1/11*c^3*e^4*x^11 + 2/5*c^3*d*e^3*x^10 + 1/3*(2*c^3*d^2*e^2 + a*c^2*e^4)*x ^9 + 2*a^3*d^3*e*x^2 + 1/2*(c^3*d^3*e + 3*a*c^2*d*e^3)*x^8 + a^3*d^4*x + 1 /7*(c^3*d^4 + 18*a*c^2*d^2*e^2 + 3*a^2*c*e^4)*x^7 + 2*(a*c^2*d^3*e + a^2*c *d*e^3)*x^6 + 1/5*(3*a*c^2*d^4 + 18*a^2*c*d^2*e^2 + a^3*e^4)*x^5 + (3*a^2* c*d^3*e + a^3*d*e^3)*x^4 + (a^2*c*d^4 + 2*a^3*d^2*e^2)*x^3
Time = 0.04 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.36 \[ \int (d+e x)^4 \left (a+c x^2\right )^3 \, dx=a^{3} d^{4} x + 2 a^{3} d^{3} e x^{2} + \frac {2 c^{3} d e^{3} x^{10}}{5} + \frac {c^{3} e^{4} x^{11}}{11} + x^{9} \left (\frac {a c^{2} e^{4}}{3} + \frac {2 c^{3} d^{2} e^{2}}{3}\right ) + x^{8} \cdot \left (\frac {3 a c^{2} d e^{3}}{2} + \frac {c^{3} d^{3} e}{2}\right ) + x^{7} \cdot \left (\frac {3 a^{2} c e^{4}}{7} + \frac {18 a c^{2} d^{2} e^{2}}{7} + \frac {c^{3} d^{4}}{7}\right ) + x^{6} \cdot \left (2 a^{2} c d e^{3} + 2 a c^{2} d^{3} e\right ) + x^{5} \left (\frac {a^{3} e^{4}}{5} + \frac {18 a^{2} c d^{2} e^{2}}{5} + \frac {3 a c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{3} d e^{3} + 3 a^{2} c d^{3} e\right ) + x^{3} \cdot \left (2 a^{3} d^{2} e^{2} + a^{2} c d^{4}\right ) \]
a**3*d**4*x + 2*a**3*d**3*e*x**2 + 2*c**3*d*e**3*x**10/5 + c**3*e**4*x**11 /11 + x**9*(a*c**2*e**4/3 + 2*c**3*d**2*e**2/3) + x**8*(3*a*c**2*d*e**3/2 + c**3*d**3*e/2) + x**7*(3*a**2*c*e**4/7 + 18*a*c**2*d**2*e**2/7 + c**3*d* *4/7) + x**6*(2*a**2*c*d*e**3 + 2*a*c**2*d**3*e) + x**5*(a**3*e**4/5 + 18* a**2*c*d**2*e**2/5 + 3*a*c**2*d**4/5) + x**4*(a**3*d*e**3 + 3*a**2*c*d**3* e) + x**3*(2*a**3*d**2*e**2 + a**2*c*d**4)
Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.23 \[ \int (d+e x)^4 \left (a+c x^2\right )^3 \, dx=\frac {1}{11} \, c^{3} e^{4} x^{11} + \frac {2}{5} \, c^{3} d e^{3} x^{10} + \frac {1}{3} \, {\left (2 \, c^{3} d^{2} e^{2} + a c^{2} e^{4}\right )} x^{9} + 2 \, a^{3} d^{3} e x^{2} + \frac {1}{2} \, {\left (c^{3} d^{3} e + 3 \, a c^{2} d e^{3}\right )} x^{8} + a^{3} d^{4} x + \frac {1}{7} \, {\left (c^{3} d^{4} + 18 \, a c^{2} d^{2} e^{2} + 3 \, a^{2} c e^{4}\right )} x^{7} + 2 \, {\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, a c^{2} d^{4} + 18 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} x^{5} + {\left (3 \, a^{2} c d^{3} e + a^{3} d e^{3}\right )} x^{4} + {\left (a^{2} c d^{4} + 2 \, a^{3} d^{2} e^{2}\right )} x^{3} \]
1/11*c^3*e^4*x^11 + 2/5*c^3*d*e^3*x^10 + 1/3*(2*c^3*d^2*e^2 + a*c^2*e^4)*x ^9 + 2*a^3*d^3*e*x^2 + 1/2*(c^3*d^3*e + 3*a*c^2*d*e^3)*x^8 + a^3*d^4*x + 1 /7*(c^3*d^4 + 18*a*c^2*d^2*e^2 + 3*a^2*c*e^4)*x^7 + 2*(a*c^2*d^3*e + a^2*c *d*e^3)*x^6 + 1/5*(3*a*c^2*d^4 + 18*a^2*c*d^2*e^2 + a^3*e^4)*x^5 + (3*a^2* c*d^3*e + a^3*d*e^3)*x^4 + (a^2*c*d^4 + 2*a^3*d^2*e^2)*x^3
Time = 0.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.31 \[ \int (d+e x)^4 \left (a+c x^2\right )^3 \, dx=\frac {1}{11} \, c^{3} e^{4} x^{11} + \frac {2}{5} \, c^{3} d e^{3} x^{10} + \frac {2}{3} \, c^{3} d^{2} e^{2} x^{9} + \frac {1}{3} \, a c^{2} e^{4} x^{9} + \frac {1}{2} \, c^{3} d^{3} e x^{8} + \frac {3}{2} \, a c^{2} d e^{3} x^{8} + \frac {1}{7} \, c^{3} d^{4} x^{7} + \frac {18}{7} \, a c^{2} d^{2} e^{2} x^{7} + \frac {3}{7} \, a^{2} c e^{4} x^{7} + 2 \, a c^{2} d^{3} e x^{6} + 2 \, a^{2} c d e^{3} x^{6} + \frac {3}{5} \, a c^{2} d^{4} x^{5} + \frac {18}{5} \, a^{2} c d^{2} e^{2} x^{5} + \frac {1}{5} \, a^{3} e^{4} x^{5} + 3 \, a^{2} c d^{3} e x^{4} + a^{3} d e^{3} x^{4} + a^{2} c d^{4} x^{3} + 2 \, a^{3} d^{2} e^{2} x^{3} + 2 \, a^{3} d^{3} e x^{2} + a^{3} d^{4} x \]
1/11*c^3*e^4*x^11 + 2/5*c^3*d*e^3*x^10 + 2/3*c^3*d^2*e^2*x^9 + 1/3*a*c^2*e ^4*x^9 + 1/2*c^3*d^3*e*x^8 + 3/2*a*c^2*d*e^3*x^8 + 1/7*c^3*d^4*x^7 + 18/7* a*c^2*d^2*e^2*x^7 + 3/7*a^2*c*e^4*x^7 + 2*a*c^2*d^3*e*x^6 + 2*a^2*c*d*e^3* x^6 + 3/5*a*c^2*d^4*x^5 + 18/5*a^2*c*d^2*e^2*x^5 + 1/5*a^3*e^4*x^5 + 3*a^2 *c*d^3*e*x^4 + a^3*d*e^3*x^4 + a^2*c*d^4*x^3 + 2*a^3*d^2*e^2*x^3 + 2*a^3*d ^3*e*x^2 + a^3*d^4*x
Time = 9.48 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.19 \[ \int (d+e x)^4 \left (a+c x^2\right )^3 \, dx=x^3\,\left (2\,a^3\,d^2\,e^2+c\,a^2\,d^4\right )+x^9\,\left (\frac {2\,c^3\,d^2\,e^2}{3}+\frac {a\,c^2\,e^4}{3}\right )+x^5\,\left (\frac {a^3\,e^4}{5}+\frac {18\,a^2\,c\,d^2\,e^2}{5}+\frac {3\,a\,c^2\,d^4}{5}\right )+x^7\,\left (\frac {3\,a^2\,c\,e^4}{7}+\frac {18\,a\,c^2\,d^2\,e^2}{7}+\frac {c^3\,d^4}{7}\right )+a^3\,d^4\,x+\frac {c^3\,e^4\,x^{11}}{11}+2\,a^3\,d^3\,e\,x^2+\frac {2\,c^3\,d\,e^3\,x^{10}}{5}+a^2\,d\,e\,x^4\,\left (3\,c\,d^2+a\,e^2\right )+\frac {c^2\,d\,e\,x^8\,\left (c\,d^2+3\,a\,e^2\right )}{2}+2\,a\,c\,d\,e\,x^6\,\left (c\,d^2+a\,e^2\right ) \]
x^3*(a^2*c*d^4 + 2*a^3*d^2*e^2) + x^9*((a*c^2*e^4)/3 + (2*c^3*d^2*e^2)/3) + x^5*((a^3*e^4)/5 + (3*a*c^2*d^4)/5 + (18*a^2*c*d^2*e^2)/5) + x^7*((c^3*d ^4)/7 + (3*a^2*c*e^4)/7 + (18*a*c^2*d^2*e^2)/7) + a^3*d^4*x + (c^3*e^4*x^1 1)/11 + 2*a^3*d^3*e*x^2 + (2*c^3*d*e^3*x^10)/5 + a^2*d*e*x^4*(a*e^2 + 3*c* d^2) + (c^2*d*e*x^8*(3*a*e^2 + c*d^2))/2 + 2*a*c*d*e*x^6*(a*e^2 + c*d^2)