Integrand size = 20, antiderivative size = 143 \[ \int \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=d x+\frac {1}{3} (10 d+e) x^3+(9 d+2 e) x^5+\frac {15}{7} (8 d+3 e) x^7+\frac {10}{3} (7 d+4 e) x^9+\frac {42}{11} (6 d+5 e) x^{11}+\frac {42}{13} (5 d+6 e) x^{13}+2 (4 d+7 e) x^{15}+\frac {15}{17} (3 d+8 e) x^{17}+\frac {5}{19} (2 d+9 e) x^{19}+\frac {1}{21} (d+10 e) x^{21}+\frac {e x^{23}}{23} \]
d*x+1/3*(10*d+e)*x^3+(9*d+2*e)*x^5+15/7*(8*d+3*e)*x^7+10/3*(7*d+4*e)*x^9+4 2/11*(6*d+5*e)*x^11+42/13*(5*d+6*e)*x^13+2*(4*d+7*e)*x^15+15/17*(3*d+8*e)* x^17+5/19*(2*d+9*e)*x^19+1/21*(d+10*e)*x^21+1/23*e*x^23
Time = 0.01 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=d x+\frac {1}{3} (10 d+e) x^3+(9 d+2 e) x^5+\frac {15}{7} (8 d+3 e) x^7+\frac {10}{3} (7 d+4 e) x^9+\frac {42}{11} (6 d+5 e) x^{11}+\frac {42}{13} (5 d+6 e) x^{13}+2 (4 d+7 e) x^{15}+\frac {15}{17} (3 d+8 e) x^{17}+\frac {5}{19} (2 d+9 e) x^{19}+\frac {1}{21} (d+10 e) x^{21}+\frac {e x^{23}}{23} \]
d*x + ((10*d + e)*x^3)/3 + (9*d + 2*e)*x^5 + (15*(8*d + 3*e)*x^7)/7 + (10* (7*d + 4*e)*x^9)/3 + (42*(6*d + 5*e)*x^11)/11 + (42*(5*d + 6*e)*x^13)/13 + 2*(4*d + 7*e)*x^15 + (15*(3*d + 8*e)*x^17)/17 + (5*(2*d + 9*e)*x^19)/19 + ((d + 10*e)*x^21)/21 + (e*x^23)/23
Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1380, 290, 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.
| \(\displaystyle \int \left (x^4+2 x^2+1\right )^5 \left (d+e x^2\right ) \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle \int \left (x^2+1\right )^{10} \left (d+e x^2\right )dx\) |
| \(\Big \downarrow \) 290 |
| \(\displaystyle \int \left (x^{20} (d+10 e)+5 x^{18} (2 d+9 e)+15 x^{16} (3 d+8 e)+30 x^{14} (4 d+7 e)+42 x^{12} (5 d+6 e)+42 x^{10} (6 d+5 e)+30 x^8 (7 d+4 e)+15 x^6 (8 d+3 e)+5 x^4 (9 d+2 e)+x^2 (10 d+e)+d+e x^{22}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{21} x^{21} (d+10 e)+\frac {5}{19} x^{19} (2 d+9 e)+\frac {15}{17} x^{17} (3 d+8 e)+2 x^{15} (4 d+7 e)+\frac {42}{13} x^{13} (5 d+6 e)+\frac {42}{11} x^{11} (6 d+5 e)+\frac {10}{3} x^9 (7 d+4 e)+\frac {15}{7} x^7 (8 d+3 e)+x^5 (9 d+2 e)+\frac {1}{3} x^3 (10 d+e)+d x+\frac {e x^{23}}{23}\) |
d*x + ((10*d + e)*x^3)/3 + (9*d + 2*e)*x^5 + (15*(8*d + 3*e)*x^7)/7 + (10* (7*d + 4*e)*x^9)/3 + (42*(6*d + 5*e)*x^11)/11 + (42*(5*d + 6*e)*x^13)/13 + 2*(4*d + 7*e)*x^15 + (15*(3*d + 8*e)*x^17)/17 + (5*(2*d + 9*e)*x^19)/19 + ((d + 10*e)*x^21)/21 + (e*x^23)/23
3.1.61.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d }, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.85
| method | result | size |
| norman | \(d x +\left (\frac {10 d}{3}+\frac {e}{3}\right ) x^{3}+\left (9 d +2 e \right ) x^{5}+\left (\frac {120 d}{7}+\frac {45 e}{7}\right ) x^{7}+\left (\frac {70 d}{3}+\frac {40 e}{3}\right ) x^{9}+\left (\frac {252 d}{11}+\frac {210 e}{11}\right ) x^{11}+\left (\frac {210 d}{13}+\frac {252 e}{13}\right ) x^{13}+\left (8 d +14 e \right ) x^{15}+\left (\frac {45 d}{17}+\frac {120 e}{17}\right ) x^{17}+\left (\frac {10 d}{19}+\frac {45 e}{19}\right ) x^{19}+\left (\frac {d}{21}+\frac {10 e}{21}\right ) x^{21}+\frac {e \,x^{23}}{23}\) | \(121\) |
| default | \(\frac {e \,x^{23}}{23}+\frac {\left (d +10 e \right ) x^{21}}{21}+\frac {\left (10 d +45 e \right ) x^{19}}{19}+\frac {\left (45 d +120 e \right ) x^{17}}{17}+\frac {\left (120 d +210 e \right ) x^{15}}{15}+\frac {\left (210 d +252 e \right ) x^{13}}{13}+\frac {\left (252 d +210 e \right ) x^{11}}{11}+\frac {\left (210 d +120 e \right ) x^{9}}{9}+\frac {\left (120 d +45 e \right ) x^{7}}{7}+\frac {\left (45 d +10 e \right ) x^{5}}{5}+\frac {\left (10 d +e \right ) x^{3}}{3}+d x\) | \(127\) |
| risch | \(\frac {1}{23} e \,x^{23}+\frac {1}{21} x^{21} d +\frac {10}{21} e \,x^{21}+\frac {10}{19} x^{19} d +\frac {45}{19} x^{19} e +\frac {45}{17} x^{17} d +\frac {120}{17} x^{17} e +8 x^{15} d +14 x^{15} e +\frac {210}{13} x^{13} d +\frac {252}{13} x^{13} e +\frac {252}{11} x^{11} d +\frac {210}{11} e \,x^{11}+\frac {70}{3} d \,x^{9}+\frac {40}{3} e \,x^{9}+\frac {120}{7} d \,x^{7}+\frac {45}{7} e \,x^{7}+9 x^{5} d +2 e \,x^{5}+\frac {10}{3} x^{3} d +\frac {1}{3} e \,x^{3}+d x\) | \(131\) |
| parallelrisch | \(\frac {1}{23} e \,x^{23}+\frac {1}{21} x^{21} d +\frac {10}{21} e \,x^{21}+\frac {10}{19} x^{19} d +\frac {45}{19} x^{19} e +\frac {45}{17} x^{17} d +\frac {120}{17} x^{17} e +8 x^{15} d +14 x^{15} e +\frac {210}{13} x^{13} d +\frac {252}{13} x^{13} e +\frac {252}{11} x^{11} d +\frac {210}{11} e \,x^{11}+\frac {70}{3} d \,x^{9}+\frac {40}{3} e \,x^{9}+\frac {120}{7} d \,x^{7}+\frac {45}{7} e \,x^{7}+9 x^{5} d +2 e \,x^{5}+\frac {10}{3} x^{3} d +\frac {1}{3} e \,x^{3}+d x\) | \(131\) |
| gosper | \(\frac {x \left (969969 e \,x^{22}+1062347 d \,x^{20}+10623470 e \,x^{20}+11741730 d \,x^{18}+52837785 e \,x^{18}+59053995 d \,x^{16}+157477320 e \,x^{16}+178474296 d \,x^{14}+312330018 e \,x^{14}+360380790 d \,x^{12}+432456948 e \,x^{12}+511085484 d \,x^{10}+425904570 e \,x^{10}+520550030 d \,x^{8}+297457160 e \,x^{8}+382444920 d \,x^{6}+143416845 e \,x^{6}+200783583 d \,x^{4}+44618574 e \,x^{4}+74364290 d \,x^{2}+7436429 e \,x^{2}+22309287 d \right )}{22309287}\) | \(134\) |
d*x+(10/3*d+1/3*e)*x^3+(9*d+2*e)*x^5+(120/7*d+45/7*e)*x^7+(70/3*d+40/3*e)* x^9+(252/11*d+210/11*e)*x^11+(210/13*d+252/13*e)*x^13+(8*d+14*e)*x^15+(45/ 17*d+120/17*e)*x^17+(10/19*d+45/19*e)*x^19+(1/21*d+10/21*e)*x^21+1/23*e*x^ 23
Time = 0.24 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.87 \[ \int \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=\frac {1}{23} \, e x^{23} + \frac {1}{21} \, {\left (d + 10 \, e\right )} x^{21} + \frac {5}{19} \, {\left (2 \, d + 9 \, e\right )} x^{19} + \frac {15}{17} \, {\left (3 \, d + 8 \, e\right )} x^{17} + 2 \, {\left (4 \, d + 7 \, e\right )} x^{15} + \frac {42}{13} \, {\left (5 \, d + 6 \, e\right )} x^{13} + \frac {42}{11} \, {\left (6 \, d + 5 \, e\right )} x^{11} + \frac {10}{3} \, {\left (7 \, d + 4 \, e\right )} x^{9} + \frac {15}{7} \, {\left (8 \, d + 3 \, e\right )} x^{7} + {\left (9 \, d + 2 \, e\right )} x^{5} + \frac {1}{3} \, {\left (10 \, d + e\right )} x^{3} + d x \]
1/23*e*x^23 + 1/21*(d + 10*e)*x^21 + 5/19*(2*d + 9*e)*x^19 + 15/17*(3*d + 8*e)*x^17 + 2*(4*d + 7*e)*x^15 + 42/13*(5*d + 6*e)*x^13 + 42/11*(6*d + 5*e )*x^11 + 10/3*(7*d + 4*e)*x^9 + 15/7*(8*d + 3*e)*x^7 + (9*d + 2*e)*x^5 + 1 /3*(10*d + e)*x^3 + d*x
Time = 0.03 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.94 \[ \int \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=d x + \frac {e x^{23}}{23} + x^{21} \left (\frac {d}{21} + \frac {10 e}{21}\right ) + x^{19} \cdot \left (\frac {10 d}{19} + \frac {45 e}{19}\right ) + x^{17} \cdot \left (\frac {45 d}{17} + \frac {120 e}{17}\right ) + x^{15} \cdot \left (8 d + 14 e\right ) + x^{13} \cdot \left (\frac {210 d}{13} + \frac {252 e}{13}\right ) + x^{11} \cdot \left (\frac {252 d}{11} + \frac {210 e}{11}\right ) + x^{9} \cdot \left (\frac {70 d}{3} + \frac {40 e}{3}\right ) + x^{7} \cdot \left (\frac {120 d}{7} + \frac {45 e}{7}\right ) + x^{5} \cdot \left (9 d + 2 e\right ) + x^{3} \cdot \left (\frac {10 d}{3} + \frac {e}{3}\right ) \]
d*x + e*x**23/23 + x**21*(d/21 + 10*e/21) + x**19*(10*d/19 + 45*e/19) + x* *17*(45*d/17 + 120*e/17) + x**15*(8*d + 14*e) + x**13*(210*d/13 + 252*e/13 ) + x**11*(252*d/11 + 210*e/11) + x**9*(70*d/3 + 40*e/3) + x**7*(120*d/7 + 45*e/7) + x**5*(9*d + 2*e) + x**3*(10*d/3 + e/3)
Time = 0.18 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.87 \[ \int \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=\frac {1}{23} \, e x^{23} + \frac {1}{21} \, {\left (d + 10 \, e\right )} x^{21} + \frac {5}{19} \, {\left (2 \, d + 9 \, e\right )} x^{19} + \frac {15}{17} \, {\left (3 \, d + 8 \, e\right )} x^{17} + 2 \, {\left (4 \, d + 7 \, e\right )} x^{15} + \frac {42}{13} \, {\left (5 \, d + 6 \, e\right )} x^{13} + \frac {42}{11} \, {\left (6 \, d + 5 \, e\right )} x^{11} + \frac {10}{3} \, {\left (7 \, d + 4 \, e\right )} x^{9} + \frac {15}{7} \, {\left (8 \, d + 3 \, e\right )} x^{7} + {\left (9 \, d + 2 \, e\right )} x^{5} + \frac {1}{3} \, {\left (10 \, d + e\right )} x^{3} + d x \]
1/23*e*x^23 + 1/21*(d + 10*e)*x^21 + 5/19*(2*d + 9*e)*x^19 + 15/17*(3*d + 8*e)*x^17 + 2*(4*d + 7*e)*x^15 + 42/13*(5*d + 6*e)*x^13 + 42/11*(6*d + 5*e )*x^11 + 10/3*(7*d + 4*e)*x^9 + 15/7*(8*d + 3*e)*x^7 + (9*d + 2*e)*x^5 + 1 /3*(10*d + e)*x^3 + d*x
Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.91 \[ \int \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=\frac {1}{23} \, e x^{23} + \frac {1}{21} \, d x^{21} + \frac {10}{21} \, e x^{21} + \frac {10}{19} \, d x^{19} + \frac {45}{19} \, e x^{19} + \frac {45}{17} \, d x^{17} + \frac {120}{17} \, e x^{17} + 8 \, d x^{15} + 14 \, e x^{15} + \frac {210}{13} \, d x^{13} + \frac {252}{13} \, e x^{13} + \frac {252}{11} \, d x^{11} + \frac {210}{11} \, e x^{11} + \frac {70}{3} \, d x^{9} + \frac {40}{3} \, e x^{9} + \frac {120}{7} \, d x^{7} + \frac {45}{7} \, e x^{7} + 9 \, d x^{5} + 2 \, e x^{5} + \frac {10}{3} \, d x^{3} + \frac {1}{3} \, e x^{3} + d x \]
1/23*e*x^23 + 1/21*d*x^21 + 10/21*e*x^21 + 10/19*d*x^19 + 45/19*e*x^19 + 4 5/17*d*x^17 + 120/17*e*x^17 + 8*d*x^15 + 14*e*x^15 + 210/13*d*x^13 + 252/1 3*e*x^13 + 252/11*d*x^11 + 210/11*e*x^11 + 70/3*d*x^9 + 40/3*e*x^9 + 120/7 *d*x^7 + 45/7*e*x^7 + 9*d*x^5 + 2*e*x^5 + 10/3*d*x^3 + 1/3*e*x^3 + d*x
Time = 0.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.84 \[ \int \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=\frac {e\,x^{23}}{23}+\left (\frac {d}{21}+\frac {10\,e}{21}\right )\,x^{21}+\left (\frac {10\,d}{19}+\frac {45\,e}{19}\right )\,x^{19}+\left (\frac {45\,d}{17}+\frac {120\,e}{17}\right )\,x^{17}+\left (8\,d+14\,e\right )\,x^{15}+\left (\frac {210\,d}{13}+\frac {252\,e}{13}\right )\,x^{13}+\left (\frac {252\,d}{11}+\frac {210\,e}{11}\right )\,x^{11}+\left (\frac {70\,d}{3}+\frac {40\,e}{3}\right )\,x^9+\left (\frac {120\,d}{7}+\frac {45\,e}{7}\right )\,x^7+\left (9\,d+2\,e\right )\,x^5+\left (\frac {10\,d}{3}+\frac {e}{3}\right )\,x^3+d\,x \]