Integrand size = 23, antiderivative size = 203 \[ \int (f x)^m \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=\frac {(f x)^{1+m}}{f (1+m)}+\frac {11 (f x)^{3+m}}{f^3 (3+m)}+\frac {55 (f x)^{5+m}}{f^5 (5+m)}+\frac {165 (f x)^{7+m}}{f^7 (7+m)}+\frac {330 (f x)^{9+m}}{f^9 (9+m)}+\frac {462 (f x)^{11+m}}{f^{11} (11+m)}+\frac {462 (f x)^{13+m}}{f^{13} (13+m)}+\frac {330 (f x)^{15+m}}{f^{15} (15+m)}+\frac {165 (f x)^{17+m}}{f^{17} (17+m)}+\frac {55 (f x)^{19+m}}{f^{19} (19+m)}+\frac {11 (f x)^{21+m}}{f^{21} (21+m)}+\frac {(f x)^{23+m}}{f^{23} (23+m)} \]
(f*x)^(1+m)/f/(1+m)+11*(f*x)^(3+m)/f^3/(3+m)+55*(f*x)^(5+m)/f^5/(5+m)+165* (f*x)^(7+m)/f^7/(7+m)+330*(f*x)^(9+m)/f^9/(9+m)+462*(f*x)^(11+m)/f^11/(11+ m)+462*(f*x)^(13+m)/f^13/(13+m)+330*(f*x)^(15+m)/f^15/(15+m)+165*(f*x)^(17 +m)/f^17/(17+m)+55*(f*x)^(19+m)/f^19/(19+m)+11*(f*x)^(21+m)/f^21/(21+m)+(f *x)^(23+m)/f^23/(23+m)
Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.60 \[ \int (f x)^m \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=x (f x)^m \left (\frac {1}{1+m}+\frac {11 x^2}{3+m}+\frac {55 x^4}{5+m}+\frac {165 x^6}{7+m}+\frac {330 x^8}{9+m}+\frac {462 x^{10}}{11+m}+\frac {462 x^{12}}{13+m}+\frac {330 x^{14}}{15+m}+\frac {165 x^{16}}{17+m}+\frac {55 x^{18}}{19+m}+\frac {11 x^{20}}{21+m}+\frac {x^{22}}{23+m}\right ) \]
x*(f*x)^m*((1 + m)^(-1) + (11*x^2)/(3 + m) + (55*x^4)/(5 + m) + (165*x^6)/ (7 + m) + (330*x^8)/(9 + m) + (462*x^10)/(11 + m) + (462*x^12)/(13 + m) + (330*x^14)/(15 + m) + (165*x^16)/(17 + m) + (55*x^18)/(19 + m) + (11*x^20) /(21 + m) + x^22/(23 + m))
Time = 0.34 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1380, 244, 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.
| \(\displaystyle \int \left (x^2+1\right ) \left (x^4+2 x^2+1\right )^5 (f x)^m \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle \int \left (x^2+1\right )^{11} (f x)^mdx\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \int \left (\frac {(f x)^{m+22}}{f^{22}}+\frac {11 (f x)^{m+20}}{f^{20}}+\frac {55 (f x)^{m+18}}{f^{18}}+\frac {165 (f x)^{m+16}}{f^{16}}+\frac {330 (f x)^{m+14}}{f^{14}}+\frac {462 (f x)^{m+12}}{f^{12}}+\frac {462 (f x)^{m+10}}{f^{10}}+\frac {330 (f x)^{m+8}}{f^8}+\frac {165 (f x)^{m+6}}{f^6}+\frac {55 (f x)^{m+4}}{f^4}+\frac {11 (f x)^{m+2}}{f^2}+(f x)^m\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(f x)^{m+23}}{f^{23} (m+23)}+\frac {11 (f x)^{m+21}}{f^{21} (m+21)}+\frac {55 (f x)^{m+19}}{f^{19} (m+19)}+\frac {165 (f x)^{m+17}}{f^{17} (m+17)}+\frac {330 (f x)^{m+15}}{f^{15} (m+15)}+\frac {462 (f x)^{m+13}}{f^{13} (m+13)}+\frac {462 (f x)^{m+11}}{f^{11} (m+11)}+\frac {330 (f x)^{m+9}}{f^9 (m+9)}+\frac {165 (f x)^{m+7}}{f^7 (m+7)}+\frac {55 (f x)^{m+5}}{f^5 (m+5)}+\frac {11 (f x)^{m+3}}{f^3 (m+3)}+\frac {(f x)^{m+1}}{f (m+1)}\) |
(f*x)^(1 + m)/(f*(1 + m)) + (11*(f*x)^(3 + m))/(f^3*(3 + m)) + (55*(f*x)^( 5 + m))/(f^5*(5 + m)) + (165*(f*x)^(7 + m))/(f^7*(7 + m)) + (330*(f*x)^(9 + m))/(f^9*(9 + m)) + (462*(f*x)^(11 + m))/(f^11*(11 + m)) + (462*(f*x)^(1 3 + m))/(f^13*(13 + m)) + (330*(f*x)^(15 + m))/(f^15*(15 + m)) + (165*(f*x )^(17 + m))/(f^17*(17 + m)) + (55*(f*x)^(19 + m))/(f^19*(19 + m)) + (11*(f *x)^(21 + m))/(f^21*(21 + m)) + (f*x)^(23 + m)/(f^23*(23 + m))
3.1.65.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1120\) vs. \(2(203)=406\).
Time = 0.65 (sec) , antiderivative size = 1121, normalized size of antiderivative = 5.52
| method | result | size |
| gosper | \(\text {Expression too large to display}\) | \(1121\) |
| risch | \(\text {Expression too large to display}\) | \(1121\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1849\) |
(f*x)^m*(m^11*x^22+121*m^10*x^22+11*m^11*x^20+6435*m^9*x^22+1353*m^10*x^20 +197835*m^8*x^22+55*m^11*x^18+72985*m^9*x^20+3889578*m^7*x^22+6875*m^10*x^ 18+2271555*m^8*x^20+51069018*m^6*x^22+165*m^11*x^16+376365*m^9*x^18+451349 58*m^7*x^20+453714470*m^5*x^22+20955*m^10*x^16+11870265*m^8*x^18+597988314 *m^6*x^20+2702025590*m^4*x^22+330*m^11*x^14+1164735*m^9*x^16+238653030*m^7 *x^18+5353566130*m^5*x^20+10431670821*m^3*x^22+42570*m^10*x^14+37263105*m^ 8*x^16+3194704590*m^6*x^18+32087153670*m^4*x^20+24372200061*m^2*x^22+462*m ^11*x^12+2403390*m^9*x^14+759091410*m^7*x^16+28857216410*m^5*x^18+12453062 6231*m^3*x^20+29985521895*m*x^22+60522*m^10*x^12+78076350*m^8*x^14+1028278 2510*m^6*x^16+174273100210*m^4*x^18+292163767533*m^2*x^20+13749310575*x^22 +462*m^11*x^10+3471930*m^9*x^12+1613983140*m^7*x^14+93862508190*m^5*x^16+6 80615362515*m^3*x^18+360568238085*m*x^20+61446*m^10*x^10+114642990*m^8*x^1 2+22164925860*m^6*x^14+572017996770*m^4*x^16+1604842704135*m^2*x^18+165646 455975*x^20+330*m^11*x^8+3582810*m^9*x^10+2408820876*m^7*x^12+204865733820 *m^5*x^14+2251106854425*m^3*x^16+1988025402825*m*x^18+44550*m^10*x^8+12036 7170*m^8*x^10+33609870756*m^6*x^12+1262375264700*m^4*x^14+5340787250535*m^ 2*x^16+915414625125*x^18+165*m^11*x^6+2640990*m^9*x^8+2575140876*m^7*x^10+ 315347150580*m^5*x^12+5015196628530*m^3*x^14+6646727085075*m*x^16+22605*m^ 10*x^6+90358290*m^8*x^8+36597992508*m^6*x^10+1969992823260*m^4*x^12+119912 58123570*m^2*x^14+3069331390125*x^16+55*m^11*x^4+1362735*m^9*x^6+197190...
Leaf count of result is larger than twice the leaf count of optimal. 759 vs. \(2 (203) = 406\).
Time = 0.26 (sec) , antiderivative size = 759, normalized size of antiderivative = 3.74 \[ \int (f x)^m \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=\text {Too large to display} \]
((m^11 + 121*m^10 + 6435*m^9 + 197835*m^8 + 3889578*m^7 + 51069018*m^6 + 4 53714470*m^5 + 2702025590*m^4 + 10431670821*m^3 + 24372200061*m^2 + 299855 21895*m + 13749310575)*x^23 + 11*(m^11 + 123*m^10 + 6635*m^9 + 206505*m^8 + 4103178*m^7 + 54362574*m^6 + 486687830*m^5 + 2917013970*m^4 + 1132096602 1*m^3 + 26560342503*m^2 + 32778930735*m + 15058768725)*x^21 + 55*(m^11 + 1 25*m^10 + 6843*m^9 + 215823*m^8 + 4339146*m^7 + 58085538*m^6 + 524676662*m ^5 + 3168601822*m^4 + 12374824773*m^3 + 29178958257*m^2 + 36145916415*m + 16643902275)*x^19 + 165*(m^11 + 127*m^10 + 7059*m^9 + 225837*m^8 + 4600554 *m^7 + 62319894*m^6 + 568863686*m^5 + 3466775738*m^4 + 13643071845*m^3 + 3 2368407579*m^2 + 40283194455*m + 18602008425)*x^17 + 330*(m^11 + 129*m^10 + 7283*m^9 + 236595*m^8 + 4890858*m^7 + 67166442*m^6 + 620805254*m^5 + 382 5379590*m^4 + 15197565541*m^3 + 36337145829*m^2 + 45488935863*m + 21082276 215)*x^15 + 462*(m^11 + 131*m^10 + 7515*m^9 + 248145*m^8 + 5213898*m^7 + 7 2748638*m^6 + 682569590*m^5 + 4264053730*m^4 + 17145560901*m^3 + 414083372 31*m^2 + 52237739295*m + 24325703325)*x^13 + 462*(m^11 + 133*m^10 + 7755*m ^9 + 260535*m^8 + 5573898*m^7 + 79216434*m^6 + 756921110*m^5 + 4811326190* m^4 + 19653671301*m^3 + 48110244633*m^2 + 61333432335*m + 28748558475)*x^1 1 + 330*(m^11 + 135*m^10 + 8003*m^9 + 273813*m^8 + 5975466*m^7 + 86750118* m^6 + 847550822*m^5 + 5509501002*m^4 + 22992750373*m^3 + 57365875587*m^2 + 74253243015*m + 35137127025)*x^9 + 165*(m^11 + 137*m^10 + 8259*m^9 + 2...
Leaf count of result is larger than twice the leaf count of optimal. 11387 vs. \(2 (177) = 354\).
Time = 2.32 (sec) , antiderivative size = 11387, normalized size of antiderivative = 56.09 \[ \int (f x)^m \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=\text {Too large to display} \]
Piecewise(((log(x) - 11/(2*x**2) - 55/(4*x**4) - 55/(2*x**6) - 165/(4*x**8 ) - 231/(5*x**10) - 77/(2*x**12) - 165/(7*x**14) - 165/(16*x**16) - 55/(18 *x**18) - 11/(20*x**20) - 1/(22*x**22))/f**23, Eq(m, -23)), ((x**2/2 + 11* log(x) - 55/(2*x**2) - 165/(4*x**4) - 55/x**6 - 231/(4*x**8) - 231/(5*x**1 0) - 55/(2*x**12) - 165/(14*x**14) - 55/(16*x**16) - 11/(18*x**18) - 1/(20 *x**20))/f**21, Eq(m, -21)), ((x**4/4 + 11*x**2/2 + 55*log(x) - 165/(2*x** 2) - 165/(2*x**4) - 77/x**6 - 231/(4*x**8) - 33/x**10 - 55/(4*x**12) - 55/ (14*x**14) - 11/(16*x**16) - 1/(18*x**18))/f**19, Eq(m, -19)), ((x**6/6 + 11*x**4/4 + 55*x**2/2 + 165*log(x) - 165/x**2 - 231/(2*x**4) - 77/x**6 - 1 65/(4*x**8) - 33/(2*x**10) - 55/(12*x**12) - 11/(14*x**14) - 1/(16*x**16)) /f**17, Eq(m, -17)), ((x**8/8 + 11*x**6/6 + 55*x**4/4 + 165*x**2/2 + 330*l og(x) - 231/x**2 - 231/(2*x**4) - 55/x**6 - 165/(8*x**8) - 11/(2*x**10) - 11/(12*x**12) - 1/(14*x**14))/f**15, Eq(m, -15)), ((x**10/10 + 11*x**8/8 + 55*x**6/6 + 165*x**4/4 + 165*x**2 + 462*log(x) - 231/x**2 - 165/(2*x**4) - 55/(2*x**6) - 55/(8*x**8) - 11/(10*x**10) - 1/(12*x**12))/f**13, Eq(m, - 13)), ((x**12/12 + 11*x**10/10 + 55*x**8/8 + 55*x**6/2 + 165*x**4/2 + 231* x**2 + 462*log(x) - 165/x**2 - 165/(4*x**4) - 55/(6*x**6) - 11/(8*x**8) - 1/(10*x**10))/f**11, Eq(m, -11)), ((x**14/14 + 11*x**12/12 + 11*x**10/2 + 165*x**8/8 + 55*x**6 + 231*x**4/2 + 231*x**2 + 330*log(x) - 165/(2*x**2) - 55/(4*x**4) - 11/(6*x**6) - 1/(8*x**8))/f**9, Eq(m, -9)), ((x**16/16 +...
Time = 0.21 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.95 \[ \int (f x)^m \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=\frac {f^{m} x^{23} x^{m}}{m + 23} + \frac {11 \, f^{m} x^{21} x^{m}}{m + 21} + \frac {55 \, f^{m} x^{19} x^{m}}{m + 19} + \frac {165 \, f^{m} x^{17} x^{m}}{m + 17} + \frac {330 \, f^{m} x^{15} x^{m}}{m + 15} + \frac {462 \, f^{m} x^{13} x^{m}}{m + 13} + \frac {462 \, f^{m} x^{11} x^{m}}{m + 11} + \frac {330 \, f^{m} x^{9} x^{m}}{m + 9} + \frac {165 \, f^{m} x^{7} x^{m}}{m + 7} + \frac {55 \, f^{m} x^{5} x^{m}}{m + 5} + \frac {11 \, f^{m} x^{3} x^{m}}{m + 3} + \frac {\left (f x\right )^{m + 1}}{f {\left (m + 1\right )}} \]
f^m*x^23*x^m/(m + 23) + 11*f^m*x^21*x^m/(m + 21) + 55*f^m*x^19*x^m/(m + 19 ) + 165*f^m*x^17*x^m/(m + 17) + 330*f^m*x^15*x^m/(m + 15) + 462*f^m*x^13*x ^m/(m + 13) + 462*f^m*x^11*x^m/(m + 11) + 330*f^m*x^9*x^m/(m + 9) + 165*f^ m*x^7*x^m/(m + 7) + 55*f^m*x^5*x^m/(m + 5) + 11*f^m*x^3*x^m/(m + 3) + (f*x )^(m + 1)/(f*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 1848 vs. \(2 (203) = 406\).
Time = 0.30 (sec) , antiderivative size = 1848, normalized size of antiderivative = 9.10 \[ \int (f x)^m \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=\text {Too large to display} \]
((f*x)^m*m^11*x^23 + 121*(f*x)^m*m^10*x^23 + 11*(f*x)^m*m^11*x^21 + 6435*( f*x)^m*m^9*x^23 + 1353*(f*x)^m*m^10*x^21 + 197835*(f*x)^m*m^8*x^23 + 55*(f *x)^m*m^11*x^19 + 72985*(f*x)^m*m^9*x^21 + 3889578*(f*x)^m*m^7*x^23 + 6875 *(f*x)^m*m^10*x^19 + 2271555*(f*x)^m*m^8*x^21 + 51069018*(f*x)^m*m^6*x^23 + 165*(f*x)^m*m^11*x^17 + 376365*(f*x)^m*m^9*x^19 + 45134958*(f*x)^m*m^7*x ^21 + 453714470*(f*x)^m*m^5*x^23 + 20955*(f*x)^m*m^10*x^17 + 11870265*(f*x )^m*m^8*x^19 + 597988314*(f*x)^m*m^6*x^21 + 2702025590*(f*x)^m*m^4*x^23 + 330*(f*x)^m*m^11*x^15 + 1164735*(f*x)^m*m^9*x^17 + 238653030*(f*x)^m*m^7*x ^19 + 5353566130*(f*x)^m*m^5*x^21 + 10431670821*(f*x)^m*m^3*x^23 + 42570*( f*x)^m*m^10*x^15 + 37263105*(f*x)^m*m^8*x^17 + 3194704590*(f*x)^m*m^6*x^19 + 32087153670*(f*x)^m*m^4*x^21 + 24372200061*(f*x)^m*m^2*x^23 + 462*(f*x) ^m*m^11*x^13 + 2403390*(f*x)^m*m^9*x^15 + 759091410*(f*x)^m*m^7*x^17 + 288 57216410*(f*x)^m*m^5*x^19 + 124530626231*(f*x)^m*m^3*x^21 + 29985521895*(f *x)^m*m*x^23 + 60522*(f*x)^m*m^10*x^13 + 78076350*(f*x)^m*m^8*x^15 + 10282 782510*(f*x)^m*m^6*x^17 + 174273100210*(f*x)^m*m^4*x^19 + 292163767533*(f* x)^m*m^2*x^21 + 13749310575*(f*x)^m*x^23 + 462*(f*x)^m*m^11*x^11 + 3471930 *(f*x)^m*m^9*x^13 + 1613983140*(f*x)^m*m^7*x^15 + 93862508190*(f*x)^m*m^5* x^17 + 680615362515*(f*x)^m*m^3*x^19 + 360568238085*(f*x)^m*m*x^21 + 61446 *(f*x)^m*m^10*x^11 + 114642990*(f*x)^m*m^8*x^13 + 22164925860*(f*x)^m*m^6* x^15 + 572017996770*(f*x)^m*m^4*x^17 + 1604842704135*(f*x)^m*m^2*x^19 +...
Time = 8.34 (sec) , antiderivative size = 1483, normalized size of antiderivative = 7.31 \[ \int (f x)^m \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx=\text {Too large to display} \]
(x^3*(f*x)^m*(2192684754645*m + 1434440867211*m^2 + 490955350391*m^3 + 102 468500970*m^4 + 14014513810*m^5 + 1298935638*m^6 + 82295598*m^7 + 3514005* m^8 + 96745*m^9 + 1551*m^10 + 11*m^11 + 1159525191825))/(703416314160*m + 590546123298*m^2 + 264300628944*m^3 + 72578259391*m^4 + 13137458400*m^5 + 1628301884*m^6 + 140529312*m^7 + 8439783*m^8 + 345840*m^9 + 9218*m^10 + 14 4*m^11 + m^12 + 316234143225) + (x^19*(f*x)^m*(1988025402825*m + 160484270 4135*m^2 + 680615362515*m^3 + 174273100210*m^4 + 28857216410*m^5 + 3194704 590*m^6 + 238653030*m^7 + 11870265*m^8 + 376365*m^9 + 6875*m^10 + 55*m^11 + 915414625125))/(703416314160*m + 590546123298*m^2 + 264300628944*m^3 + 7 2578259391*m^4 + 13137458400*m^5 + 1628301884*m^6 + 140529312*m^7 + 843978 3*m^8 + 345840*m^9 + 9218*m^10 + 144*m^11 + m^12 + 316234143225) + (x^11*( f*x)^m*(28336045738770*m + 22226933020446*m^2 + 9079996141062*m^3 + 222283 2699780*m^4 + 349697552820*m^5 + 36597992508*m^6 + 2575140876*m^7 + 120367 170*m^8 + 3582810*m^9 + 61446*m^10 + 462*m^11 + 13281834015450))/(70341631 4160*m + 590546123298*m^2 + 264300628944*m^3 + 72578259391*m^4 + 131374584 00*m^5 + 1628301884*m^6 + 140529312*m^7 + 8439783*m^8 + 345840*m^9 + 9218* m^10 + 144*m^11 + m^12 + 316234143225) + (x^21*(f*x)^m*(360568238085*m + 2 92163767533*m^2 + 124530626231*m^3 + 32087153670*m^4 + 5353566130*m^5 + 59 7988314*m^6 + 45134958*m^7 + 2271555*m^8 + 72985*m^9 + 1353*m^10 + 11*m^11 + 165646455975))/(703416314160*m + 590546123298*m^2 + 264300628944*m^3...