Integrand size = 20, antiderivative size = 149 \[ \int x^2 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {1}{3} a^5 d x^3+\frac {1}{5} a^5 e x^5+\frac {5}{7} a^4 c d x^7+\frac {5}{9} a^4 c e x^9+\frac {10}{11} a^3 c^2 d x^{11}+\frac {10}{13} a^3 c^2 e x^{13}+\frac {2}{3} a^2 c^3 d x^{15}+\frac {10}{17} a^2 c^3 e x^{17}+\frac {5}{19} a c^4 d x^{19}+\frac {5}{21} a c^4 e x^{21}+\frac {1}{23} c^5 d x^{23}+\frac {1}{25} c^5 e x^{25} \]
1/3*a^5*d*x^3+1/5*a^5*e*x^5+5/7*a^4*c*d*x^7+5/9*a^4*c*e*x^9+10/11*a^3*c^2* d*x^11+10/13*a^3*c^2*e*x^13+2/3*a^2*c^3*d*x^15+10/17*a^2*c^3*e*x^17+5/19*a *c^4*d*x^19+5/21*a*c^4*e*x^21+1/23*c^5*d*x^23+1/25*c^5*e*x^25
Time = 0.00 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00 \[ \int x^2 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {1}{3} a^5 d x^3+\frac {1}{5} a^5 e x^5+\frac {5}{7} a^4 c d x^7+\frac {5}{9} a^4 c e x^9+\frac {10}{11} a^3 c^2 d x^{11}+\frac {10}{13} a^3 c^2 e x^{13}+\frac {2}{3} a^2 c^3 d x^{15}+\frac {10}{17} a^2 c^3 e x^{17}+\frac {5}{19} a c^4 d x^{19}+\frac {5}{21} a c^4 e x^{21}+\frac {1}{23} c^5 d x^{23}+\frac {1}{25} c^5 e x^{25} \]
(a^5*d*x^3)/3 + (a^5*e*x^5)/5 + (5*a^4*c*d*x^7)/7 + (5*a^4*c*e*x^9)/9 + (1 0*a^3*c^2*d*x^11)/11 + (10*a^3*c^2*e*x^13)/13 + (2*a^2*c^3*d*x^15)/3 + (10 *a^2*c^3*e*x^17)/17 + (5*a*c^4*d*x^19)/19 + (5*a*c^4*e*x^21)/21 + (c^5*d*x ^23)/23 + (c^5*e*x^25)/25
Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1585, 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.
\(\displaystyle \int x^2 \left (a+c x^4\right )^5 \left (d+e x^2\right ) \, dx\) |
\(\Big \downarrow \) 1585 |
\(\displaystyle \int \left (a^5 d x^2+a^5 e x^4+5 a^4 c d x^6+5 a^4 c e x^8+10 a^3 c^2 d x^{10}+10 a^3 c^2 e x^{12}+10 a^2 c^3 d x^{14}+10 a^2 c^3 e x^{16}+5 a c^4 d x^{18}+5 a c^4 e x^{20}+c^5 d x^{22}+c^5 e x^{24}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} a^5 d x^3+\frac {1}{5} a^5 e x^5+\frac {5}{7} a^4 c d x^7+\frac {5}{9} a^4 c e x^9+\frac {10}{11} a^3 c^2 d x^{11}+\frac {10}{13} a^3 c^2 e x^{13}+\frac {2}{3} a^2 c^3 d x^{15}+\frac {10}{17} a^2 c^3 e x^{17}+\frac {5}{19} a c^4 d x^{19}+\frac {5}{21} a c^4 e x^{21}+\frac {1}{23} c^5 d x^{23}+\frac {1}{25} c^5 e x^{25}\) |
(a^5*d*x^3)/3 + (a^5*e*x^5)/5 + (5*a^4*c*d*x^7)/7 + (5*a^4*c*e*x^9)/9 + (1 0*a^3*c^2*d*x^11)/11 + (10*a^3*c^2*e*x^13)/13 + (2*a^2*c^3*d*x^15)/3 + (10 *a^2*c^3*e*x^17)/17 + (5*a*c^4*d*x^19)/19 + (5*a*c^4*e*x^21)/21 + (c^5*d*x ^23)/23 + (c^5*e*x^25)/25
3.1.2.3.1 Defintions of rubi rules used
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && IGtQ[p, 0] && IGtQ[q, -2]
Time = 0.29 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(\frac {1}{3} a^{5} d \,x^{3}+\frac {1}{5} a^{5} e \,x^{5}+\frac {5}{7} a^{4} c d \,x^{7}+\frac {5}{9} a^{4} c e \,x^{9}+\frac {10}{11} a^{3} c^{2} d \,x^{11}+\frac {10}{13} a^{3} c^{2} e \,x^{13}+\frac {2}{3} a^{2} c^{3} d \,x^{15}+\frac {10}{17} a^{2} c^{3} e \,x^{17}+\frac {5}{19} a \,c^{4} d \,x^{19}+\frac {5}{21} a \,c^{4} e \,x^{21}+\frac {1}{23} c^{5} d \,x^{23}+\frac {1}{25} c^{5} e \,x^{25}\) | \(126\) |
default | \(\frac {1}{3} a^{5} d \,x^{3}+\frac {1}{5} a^{5} e \,x^{5}+\frac {5}{7} a^{4} c d \,x^{7}+\frac {5}{9} a^{4} c e \,x^{9}+\frac {10}{11} a^{3} c^{2} d \,x^{11}+\frac {10}{13} a^{3} c^{2} e \,x^{13}+\frac {2}{3} a^{2} c^{3} d \,x^{15}+\frac {10}{17} a^{2} c^{3} e \,x^{17}+\frac {5}{19} a \,c^{4} d \,x^{19}+\frac {5}{21} a \,c^{4} e \,x^{21}+\frac {1}{23} c^{5} d \,x^{23}+\frac {1}{25} c^{5} e \,x^{25}\) | \(126\) |
norman | \(\frac {1}{3} a^{5} d \,x^{3}+\frac {1}{5} a^{5} e \,x^{5}+\frac {5}{7} a^{4} c d \,x^{7}+\frac {5}{9} a^{4} c e \,x^{9}+\frac {10}{11} a^{3} c^{2} d \,x^{11}+\frac {10}{13} a^{3} c^{2} e \,x^{13}+\frac {2}{3} a^{2} c^{3} d \,x^{15}+\frac {10}{17} a^{2} c^{3} e \,x^{17}+\frac {5}{19} a \,c^{4} d \,x^{19}+\frac {5}{21} a \,c^{4} e \,x^{21}+\frac {1}{23} c^{5} d \,x^{23}+\frac {1}{25} c^{5} e \,x^{25}\) | \(126\) |
risch | \(\frac {1}{3} a^{5} d \,x^{3}+\frac {1}{5} a^{5} e \,x^{5}+\frac {5}{7} a^{4} c d \,x^{7}+\frac {5}{9} a^{4} c e \,x^{9}+\frac {10}{11} a^{3} c^{2} d \,x^{11}+\frac {10}{13} a^{3} c^{2} e \,x^{13}+\frac {2}{3} a^{2} c^{3} d \,x^{15}+\frac {10}{17} a^{2} c^{3} e \,x^{17}+\frac {5}{19} a \,c^{4} d \,x^{19}+\frac {5}{21} a \,c^{4} e \,x^{21}+\frac {1}{23} c^{5} d \,x^{23}+\frac {1}{25} c^{5} e \,x^{25}\) | \(126\) |
parallelrisch | \(\frac {1}{3} a^{5} d \,x^{3}+\frac {1}{5} a^{5} e \,x^{5}+\frac {5}{7} a^{4} c d \,x^{7}+\frac {5}{9} a^{4} c e \,x^{9}+\frac {10}{11} a^{3} c^{2} d \,x^{11}+\frac {10}{13} a^{3} c^{2} e \,x^{13}+\frac {2}{3} a^{2} c^{3} d \,x^{15}+\frac {10}{17} a^{2} c^{3} e \,x^{17}+\frac {5}{19} a \,c^{4} d \,x^{19}+\frac {5}{21} a \,c^{4} e \,x^{21}+\frac {1}{23} c^{5} d \,x^{23}+\frac {1}{25} c^{5} e \,x^{25}\) | \(126\) |
1/3*a^5*d*x^3+1/5*a^5*e*x^5+5/7*a^4*c*d*x^7+5/9*a^4*c*e*x^9+10/11*a^3*c^2* d*x^11+10/13*a^3*c^2*e*x^13+2/3*a^2*c^3*d*x^15+10/17*a^2*c^3*e*x^17+5/19*a *c^4*d*x^19+5/21*a*c^4*e*x^21+1/23*c^5*d*x^23+1/25*c^5*e*x^25
Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84 \[ \int x^2 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {1}{25} \, c^{5} e x^{25} + \frac {1}{23} \, c^{5} d x^{23} + \frac {5}{21} \, a c^{4} e x^{21} + \frac {5}{19} \, a c^{4} d x^{19} + \frac {10}{17} \, a^{2} c^{3} e x^{17} + \frac {2}{3} \, a^{2} c^{3} d x^{15} + \frac {10}{13} \, a^{3} c^{2} e x^{13} + \frac {10}{11} \, a^{3} c^{2} d x^{11} + \frac {5}{9} \, a^{4} c e x^{9} + \frac {5}{7} \, a^{4} c d x^{7} + \frac {1}{5} \, a^{5} e x^{5} + \frac {1}{3} \, a^{5} d x^{3} \]
1/25*c^5*e*x^25 + 1/23*c^5*d*x^23 + 5/21*a*c^4*e*x^21 + 5/19*a*c^4*d*x^19 + 10/17*a^2*c^3*e*x^17 + 2/3*a^2*c^3*d*x^15 + 10/13*a^3*c^2*e*x^13 + 10/11 *a^3*c^2*d*x^11 + 5/9*a^4*c*e*x^9 + 5/7*a^4*c*d*x^7 + 1/5*a^5*e*x^5 + 1/3* a^5*d*x^3
Time = 0.02 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.04 \[ \int x^2 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {a^{5} d x^{3}}{3} + \frac {a^{5} e x^{5}}{5} + \frac {5 a^{4} c d x^{7}}{7} + \frac {5 a^{4} c e x^{9}}{9} + \frac {10 a^{3} c^{2} d x^{11}}{11} + \frac {10 a^{3} c^{2} e x^{13}}{13} + \frac {2 a^{2} c^{3} d x^{15}}{3} + \frac {10 a^{2} c^{3} e x^{17}}{17} + \frac {5 a c^{4} d x^{19}}{19} + \frac {5 a c^{4} e x^{21}}{21} + \frac {c^{5} d x^{23}}{23} + \frac {c^{5} e x^{25}}{25} \]
a**5*d*x**3/3 + a**5*e*x**5/5 + 5*a**4*c*d*x**7/7 + 5*a**4*c*e*x**9/9 + 10 *a**3*c**2*d*x**11/11 + 10*a**3*c**2*e*x**13/13 + 2*a**2*c**3*d*x**15/3 + 10*a**2*c**3*e*x**17/17 + 5*a*c**4*d*x**19/19 + 5*a*c**4*e*x**21/21 + c**5 *d*x**23/23 + c**5*e*x**25/25
Time = 0.20 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84 \[ \int x^2 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {1}{25} \, c^{5} e x^{25} + \frac {1}{23} \, c^{5} d x^{23} + \frac {5}{21} \, a c^{4} e x^{21} + \frac {5}{19} \, a c^{4} d x^{19} + \frac {10}{17} \, a^{2} c^{3} e x^{17} + \frac {2}{3} \, a^{2} c^{3} d x^{15} + \frac {10}{13} \, a^{3} c^{2} e x^{13} + \frac {10}{11} \, a^{3} c^{2} d x^{11} + \frac {5}{9} \, a^{4} c e x^{9} + \frac {5}{7} \, a^{4} c d x^{7} + \frac {1}{5} \, a^{5} e x^{5} + \frac {1}{3} \, a^{5} d x^{3} \]
1/25*c^5*e*x^25 + 1/23*c^5*d*x^23 + 5/21*a*c^4*e*x^21 + 5/19*a*c^4*d*x^19 + 10/17*a^2*c^3*e*x^17 + 2/3*a^2*c^3*d*x^15 + 10/13*a^3*c^2*e*x^13 + 10/11 *a^3*c^2*d*x^11 + 5/9*a^4*c*e*x^9 + 5/7*a^4*c*d*x^7 + 1/5*a^5*e*x^5 + 1/3* a^5*d*x^3
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84 \[ \int x^2 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {1}{25} \, c^{5} e x^{25} + \frac {1}{23} \, c^{5} d x^{23} + \frac {5}{21} \, a c^{4} e x^{21} + \frac {5}{19} \, a c^{4} d x^{19} + \frac {10}{17} \, a^{2} c^{3} e x^{17} + \frac {2}{3} \, a^{2} c^{3} d x^{15} + \frac {10}{13} \, a^{3} c^{2} e x^{13} + \frac {10}{11} \, a^{3} c^{2} d x^{11} + \frac {5}{9} \, a^{4} c e x^{9} + \frac {5}{7} \, a^{4} c d x^{7} + \frac {1}{5} \, a^{5} e x^{5} + \frac {1}{3} \, a^{5} d x^{3} \]
1/25*c^5*e*x^25 + 1/23*c^5*d*x^23 + 5/21*a*c^4*e*x^21 + 5/19*a*c^4*d*x^19 + 10/17*a^2*c^3*e*x^17 + 2/3*a^2*c^3*d*x^15 + 10/13*a^3*c^2*e*x^13 + 10/11 *a^3*c^2*d*x^11 + 5/9*a^4*c*e*x^9 + 5/7*a^4*c*d*x^7 + 1/5*a^5*e*x^5 + 1/3* a^5*d*x^3
Time = 0.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.84 \[ \int x^2 \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx=\frac {e\,a^5\,x^5}{5}+\frac {d\,a^5\,x^3}{3}+\frac {5\,e\,a^4\,c\,x^9}{9}+\frac {5\,d\,a^4\,c\,x^7}{7}+\frac {10\,e\,a^3\,c^2\,x^{13}}{13}+\frac {10\,d\,a^3\,c^2\,x^{11}}{11}+\frac {10\,e\,a^2\,c^3\,x^{17}}{17}+\frac {2\,d\,a^2\,c^3\,x^{15}}{3}+\frac {5\,e\,a\,c^4\,x^{21}}{21}+\frac {5\,d\,a\,c^4\,x^{19}}{19}+\frac {e\,c^5\,x^{25}}{25}+\frac {d\,c^5\,x^{23}}{23} \]