Integrand size = 19, antiderivative size = 97 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=a^2 d^2 x+\frac {2}{3} a^2 d e x^3+\frac {1}{5} a \left (2 c d^2+a e^2\right ) x^5+\frac {4}{7} a c d e x^7+\frac {1}{9} c \left (c d^2+2 a e^2\right ) x^9+\frac {2}{11} c^2 d e x^{11}+\frac {1}{13} c^2 e^2 x^{13} \] Output:
a^2*d^2*x+2/3*a^2*d*e*x^3+1/5*a*(a*e^2+2*c*d^2)*x^5+4/7*a*c*d*e*x^7+1/9*c* (2*a*e^2+c*d^2)*x^9+2/11*c^2*d*e*x^11+1/13*c^2*e^2*x^13
Time = 0.01 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=a^2 d^2 x+\frac {2}{3} a^2 d e x^3+\frac {1}{5} a \left (2 c d^2+a e^2\right ) x^5+\frac {4}{7} a c d e x^7+\frac {1}{9} c \left (c d^2+2 a e^2\right ) x^9+\frac {2}{11} c^2 d e x^{11}+\frac {1}{13} c^2 e^2 x^{13} \] Input:
Integrate[(d + e*x^2)^2*(a + c*x^4)^2,x]
Output:
a^2*d^2*x + (2*a^2*d*e*x^3)/3 + (a*(2*c*d^2 + a*e^2)*x^5)/5 + (4*a*c*d*e*x ^7)/7 + (c*(c*d^2 + 2*a*e^2)*x^9)/9 + (2*c^2*d*e*x^11)/11 + (c^2*e^2*x^13) /13
Time = 0.44 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1468, 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.
\(\displaystyle \int \left (a+c x^4\right )^2 \left (d+e x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 1468 |
\(\displaystyle \int \left (a^2 d^2+2 a^2 d e x^2+c x^8 \left (2 a e^2+c d^2\right )+a x^4 \left (a e^2+2 c d^2\right )+4 a c d e x^6+2 c^2 d e x^{10}+c^2 e^2 x^{12}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 d^2 x+\frac {2}{3} a^2 d e x^3+\frac {1}{9} c x^9 \left (2 a e^2+c d^2\right )+\frac {1}{5} a x^5 \left (a e^2+2 c d^2\right )+\frac {4}{7} a c d e x^7+\frac {2}{11} c^2 d e x^{11}+\frac {1}{13} c^2 e^2 x^{13}\) |
Input:
Int[(d + e*x^2)^2*(a + c*x^4)^2,x]
Output:
a^2*d^2*x + (2*a^2*d*e*x^3)/3 + (a*(2*c*d^2 + a*e^2)*x^5)/5 + (4*a*c*d*e*x ^7)/7 + (c*(c*d^2 + 2*a*e^2)*x^9)/9 + (2*c^2*d*e*x^11)/11 + (c^2*e^2*x^13) /13
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> I nt[ExpandIntegrand[(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e }, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {c^{2} e^{2} x^{13}}{13}+\frac {2 c^{2} d e \,x^{11}}{11}+\frac {\left (2 e^{2} a c +d^{2} c^{2}\right ) x^{9}}{9}+\frac {4 a c d e \,x^{7}}{7}+\frac {\left (a^{2} e^{2}+2 d^{2} a c \right ) x^{5}}{5}+\frac {2 a^{2} d e \,x^{3}}{3}+a^{2} d^{2} x\) | \(90\) |
norman | \(\frac {c^{2} e^{2} x^{13}}{13}+\frac {2 c^{2} d e \,x^{11}}{11}+\left (\frac {2}{9} e^{2} a c +\frac {1}{9} d^{2} c^{2}\right ) x^{9}+\frac {4 a c d e \,x^{7}}{7}+\left (\frac {1}{5} a^{2} e^{2}+\frac {2}{5} d^{2} a c \right ) x^{5}+\frac {2 a^{2} d e \,x^{3}}{3}+a^{2} d^{2} x\) | \(90\) |
gosper | \(\frac {1}{13} c^{2} e^{2} x^{13}+\frac {2}{11} c^{2} d e \,x^{11}+\frac {2}{9} x^{9} e^{2} a c +\frac {1}{9} x^{9} d^{2} c^{2}+\frac {4}{7} a c d e \,x^{7}+\frac {1}{5} x^{5} a^{2} e^{2}+\frac {2}{5} x^{5} d^{2} a c +\frac {2}{3} a^{2} d e \,x^{3}+a^{2} d^{2} x\) | \(92\) |
risch | \(\frac {1}{13} c^{2} e^{2} x^{13}+\frac {2}{11} c^{2} d e \,x^{11}+\frac {2}{9} x^{9} e^{2} a c +\frac {1}{9} x^{9} d^{2} c^{2}+\frac {4}{7} a c d e \,x^{7}+\frac {1}{5} x^{5} a^{2} e^{2}+\frac {2}{5} x^{5} d^{2} a c +\frac {2}{3} a^{2} d e \,x^{3}+a^{2} d^{2} x\) | \(92\) |
parallelrisch | \(\frac {1}{13} c^{2} e^{2} x^{13}+\frac {2}{11} c^{2} d e \,x^{11}+\frac {2}{9} x^{9} e^{2} a c +\frac {1}{9} x^{9} d^{2} c^{2}+\frac {4}{7} a c d e \,x^{7}+\frac {1}{5} x^{5} a^{2} e^{2}+\frac {2}{5} x^{5} d^{2} a c +\frac {2}{3} a^{2} d e \,x^{3}+a^{2} d^{2} x\) | \(92\) |
orering | \(\frac {x \left (3465 e^{2} c^{2} x^{12}+8190 d e \,c^{2} x^{10}+10010 a c \,e^{2} x^{8}+5005 c^{2} d^{2} x^{8}+25740 d e a c \,x^{6}+9009 a^{2} e^{2} x^{4}+18018 a c \,d^{2} x^{4}+30030 d e \,a^{2} x^{2}+45045 a^{2} d^{2}\right )}{45045}\) | \(95\) |
Input:
int((e*x^2+d)^2*(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
1/13*c^2*e^2*x^13+2/11*c^2*d*e*x^11+1/9*(2*a*c*e^2+c^2*d^2)*x^9+4/7*a*c*d* e*x^7+1/5*(a^2*e^2+2*a*c*d^2)*x^5+2/3*a^2*d*e*x^3+a^2*d^2*x
Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.92 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=\frac {1}{13} \, c^{2} e^{2} x^{13} + \frac {2}{11} \, c^{2} d e x^{11} + \frac {4}{7} \, a c d e x^{7} + \frac {1}{9} \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{9} + \frac {2}{3} \, a^{2} d e x^{3} + \frac {1}{5} \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{5} + a^{2} d^{2} x \] Input:
integrate((e*x^2+d)^2*(c*x^4+a)^2,x, algorithm="fricas")
Output:
1/13*c^2*e^2*x^13 + 2/11*c^2*d*e*x^11 + 4/7*a*c*d*e*x^7 + 1/9*(c^2*d^2 + 2 *a*c*e^2)*x^9 + 2/3*a^2*d*e*x^3 + 1/5*(2*a*c*d^2 + a^2*e^2)*x^5 + a^2*d^2* x
Time = 0.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.07 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=a^{2} d^{2} x + \frac {2 a^{2} d e x^{3}}{3} + \frac {4 a c d e x^{7}}{7} + \frac {2 c^{2} d e x^{11}}{11} + \frac {c^{2} e^{2} x^{13}}{13} + x^{9} \cdot \left (\frac {2 a c e^{2}}{9} + \frac {c^{2} d^{2}}{9}\right ) + x^{5} \left (\frac {a^{2} e^{2}}{5} + \frac {2 a c d^{2}}{5}\right ) \] Input:
integrate((e*x**2+d)**2*(c*x**4+a)**2,x)
Output:
a**2*d**2*x + 2*a**2*d*e*x**3/3 + 4*a*c*d*e*x**7/7 + 2*c**2*d*e*x**11/11 + c**2*e**2*x**13/13 + x**9*(2*a*c*e**2/9 + c**2*d**2/9) + x**5*(a**2*e**2/ 5 + 2*a*c*d**2/5)
Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.92 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=\frac {1}{13} \, c^{2} e^{2} x^{13} + \frac {2}{11} \, c^{2} d e x^{11} + \frac {4}{7} \, a c d e x^{7} + \frac {1}{9} \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{9} + \frac {2}{3} \, a^{2} d e x^{3} + \frac {1}{5} \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{5} + a^{2} d^{2} x \] Input:
integrate((e*x^2+d)^2*(c*x^4+a)^2,x, algorithm="maxima")
Output:
1/13*c^2*e^2*x^13 + 2/11*c^2*d*e*x^11 + 4/7*a*c*d*e*x^7 + 1/9*(c^2*d^2 + 2 *a*c*e^2)*x^9 + 2/3*a^2*d*e*x^3 + 1/5*(2*a*c*d^2 + a^2*e^2)*x^5 + a^2*d^2* x
Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=\frac {1}{13} \, c^{2} e^{2} x^{13} + \frac {2}{11} \, c^{2} d e x^{11} + \frac {1}{9} \, c^{2} d^{2} x^{9} + \frac {2}{9} \, a c e^{2} x^{9} + \frac {4}{7} \, a c d e x^{7} + \frac {2}{5} \, a c d^{2} x^{5} + \frac {1}{5} \, a^{2} e^{2} x^{5} + \frac {2}{3} \, a^{2} d e x^{3} + a^{2} d^{2} x \] Input:
integrate((e*x^2+d)^2*(c*x^4+a)^2,x, algorithm="giac")
Output:
1/13*c^2*e^2*x^13 + 2/11*c^2*d*e*x^11 + 1/9*c^2*d^2*x^9 + 2/9*a*c*e^2*x^9 + 4/7*a*c*d*e*x^7 + 2/5*a*c*d^2*x^5 + 1/5*a^2*e^2*x^5 + 2/3*a^2*d*e*x^3 + a^2*d^2*x
Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.92 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=x^5\,\left (\frac {a^2\,e^2}{5}+\frac {2\,c\,a\,d^2}{5}\right )+x^9\,\left (\frac {c^2\,d^2}{9}+\frac {2\,a\,c\,e^2}{9}\right )+a^2\,d^2\,x+\frac {c^2\,e^2\,x^{13}}{13}+\frac {2\,a^2\,d\,e\,x^3}{3}+\frac {2\,c^2\,d\,e\,x^{11}}{11}+\frac {4\,a\,c\,d\,e\,x^7}{7} \] Input:
int((a + c*x^4)^2*(d + e*x^2)^2,x)
Output:
x^5*((a^2*e^2)/5 + (2*a*c*d^2)/5) + x^9*((c^2*d^2)/9 + (2*a*c*e^2)/9) + a^ 2*d^2*x + (c^2*e^2*x^13)/13 + (2*a^2*d*e*x^3)/3 + (2*c^2*d*e*x^11)/11 + (4 *a*c*d*e*x^7)/7
Time = 0.18 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \left (d+e x^2\right )^2 \left (a+c x^4\right )^2 \, dx=\frac {x \left (3465 c^{2} e^{2} x^{12}+8190 c^{2} d e \,x^{10}+10010 a c \,e^{2} x^{8}+5005 c^{2} d^{2} x^{8}+25740 a c d e \,x^{6}+9009 a^{2} e^{2} x^{4}+18018 a c \,d^{2} x^{4}+30030 a^{2} d e \,x^{2}+45045 a^{2} d^{2}\right )}{45045} \] Input:
int((e*x^2+d)^2*(c*x^4+a)^2,x)
Output:
(x*(45045*a**2*d**2 + 30030*a**2*d*e*x**2 + 9009*a**2*e**2*x**4 + 18018*a* c*d**2*x**4 + 25740*a*c*d*e*x**6 + 10010*a*c*e**2*x**8 + 5005*c**2*d**2*x* *8 + 8190*c**2*d*e*x**10 + 3465*c**2*e**2*x**12))/45045