3.65 \(\int (f x)^m \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx\)
Optimal. Leaf size=203 \[ \frac{(f x)^{m+23}}{f^{23} (m+23)}+\frac{11 (f x)^{m+21}}{f^{21} (m+21)}+\frac{55 (f x)^{m+19}}{f^{19} (m+19)}+\frac{165 (f x)^{m+17}}{f^{17} (m+17)}+\frac{330 (f x)^{m+15}}{f^{15} (m+15)}+\frac{462 (f x)^{m+13}}{f^{13} (m+13)}+\frac{462 (f x)^{m+11}}{f^{11} (m+11)}+\frac{330 (f x)^{m+9}}{f^9 (m+9)}+\frac{165 (f x)^{m+7}}{f^7 (m+7)}+\frac{55 (f x)^{m+5}}{f^5 (m+5)}+\frac{11 (f x)^{m+3}}{f^3 (m+3)}+\frac{(f x)^{m+1}}{f (m+1)} \]
[Out]
(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))
_______________________________________________________________________________________
Rubi [A] time = 0.169792, antiderivative size = 203, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087
\[ \frac{(f x)^{m+23}}{f^{23} (m+23)}+\frac{11 (f x)^{m+21}}{f^{21} (m+21)}+\frac{55 (f x)^{m+19}}{f^{19} (m+19)}+\frac{165 (f x)^{m+17}}{f^{17} (m+17)}+\frac{330 (f x)^{m+15}}{f^{15} (m+15)}+\frac{462 (f x)^{m+13}}{f^{13} (m+13)}+\frac{462 (f x)^{m+11}}{f^{11} (m+11)}+\frac{330 (f x)^{m+9}}{f^9 (m+9)}+\frac{165 (f x)^{m+7}}{f^7 (m+7)}+\frac{55 (f x)^{m+5}}{f^5 (m+5)}+\frac{11 (f x)^{m+3}}{f^3 (m+3)}+\frac{(f x)^{m+1}}{f (m+1)} \]
Antiderivative was successfully verified.
[In] Int[(f*x)^m*(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]
[Out]
(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))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 35.597, size = 177, normalized size = 0.87 \[ \frac{\left (f x\right )^{m + 1}}{f \left (m + 1\right )} + \frac{11 \left (f x\right )^{m + 3}}{f^{3} \left (m + 3\right )} + \frac{55 \left (f x\right )^{m + 5}}{f^{5} \left (m + 5\right )} + \frac{165 \left (f x\right )^{m + 7}}{f^{7} \left (m + 7\right )} + \frac{330 \left (f x\right )^{m + 9}}{f^{9} \left (m + 9\right )} + \frac{462 \left (f x\right )^{m + 11}}{f^{11} \left (m + 11\right )} + \frac{462 \left (f x\right )^{m + 13}}{f^{13} \left (m + 13\right )} + \frac{330 \left (f x\right )^{m + 15}}{f^{15} \left (m + 15\right )} + \frac{165 \left (f x\right )^{m + 17}}{f^{17} \left (m + 17\right )} + \frac{55 \left (f x\right )^{m + 19}}{f^{19} \left (m + 19\right )} + \frac{11 \left (f x\right )^{m + 21}}{f^{21} \left (m + 21\right )} + \frac{\left (f x\right )^{m + 23}}{f^{23} \left (m + 23\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x)**m*(x**2+1)*(x**4+2*x**2+1)**5,x)
[Out]
(f*x)**(m + 1)/(f*(m + 1)) + 11*(f*x)**(m + 3)/(f**3*(m + 3)) + 55*(f*x)**(m + 5
)/(f**5*(m + 5)) + 165*(f*x)**(m + 7)/(f**7*(m + 7)) + 330*(f*x)**(m + 9)/(f**9*
(m + 9)) + 462*(f*x)**(m + 11)/(f**11*(m + 11)) + 462*(f*x)**(m + 13)/(f**13*(m
+ 13)) + 330*(f*x)**(m + 15)/(f**15*(m + 15)) + 165*(f*x)**(m + 17)/(f**17*(m +
17)) + 55*(f*x)**(m + 19)/(f**19*(m + 19)) + 11*(f*x)**(m + 21)/(f**21*(m + 21))
+ (f*x)**(m + 23)/(f**23*(m + 23))
_______________________________________________________________________________________
Mathematica [A] time = 0.0569759, size = 123, normalized size = 0.61 \[ \left (\frac{x^{23}}{m+23}+\frac{11 x^{21}}{m+21}+\frac{55 x^{19}}{m+19}+\frac{165 x^{17}}{m+17}+\frac{330 x^{15}}{m+15}+\frac{462 x^{13}}{m+13}+\frac{462 x^{11}}{m+11}+\frac{330 x^9}{m+9}+\frac{165 x^7}{m+7}+\frac{55 x^5}{m+5}+\frac{11 x^3}{m+3}+\frac{x}{m+1}\right ) (f x)^m \]
Antiderivative was successfully verified.
[In] Integrate[(f*x)^m*(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]
[Out]
(f*x)^m*(x/(1 + m) + (11*x^3)/(3 + m) + (55*x^5)/(5 + m) + (165*x^7)/(7 + m) + (
330*x^9)/(9 + m) + (462*x^11)/(11 + m) + (462*x^13)/(13 + m) + (330*x^15)/(15 +
m) + (165*x^17)/(17 + m) + (55*x^19)/(19 + m) + (11*x^21)/(21 + m) + x^23/(23 +
m))
_______________________________________________________________________________________
Maple [B] time = 0.015, size = 1121, normalized size = 5.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x)^m*(x^2+1)*(x^4+2*x^2+1)^5,x)
[Out]
(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+19783
5*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+45134958*m^7*x^20+453714
470*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+1043
1670821*m^3*x^22+42570*m^10*x^14+37263105*m^8*x^16+3194704590*m^6*x^18+320871536
70*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+124530626231*m^3*x^20+29985521895*m*x^22+60522*m^10*x^12
+78076350*m^8*x^14+10282782510*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+93862508
190*m^5*x^16+680615362515*m^3*x^18+360568238085*m*x^20+61446*m^10*x^10+114642990
*m^8*x^12+22164925860*m^6*x^14+572017996770*m^4*x^16+1604842704135*m^2*x^18+1656
46455975*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+120367170*m^8*x
^10+33609870756*m^6*x^12+1262375264700*m^4*x^14+5340787250535*m^2*x^16+915414625
125*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+3659
7992508*m^6*x^10+1969992823260*m^4*x^12+11991258123570*m^2*x^14+3069331390125*x^
16+55*m^11*x^4+1362735*m^9*x^6+1971903780*m^7*x^8+349697552820*m^5*x^10+79212491
36262*m^3*x^12+15011348834790*m*x^14+7645*m^10*x^4+47524455*m^8*x^6+28627538940*
m^6*x^8+2222832699780*m^4*x^10+19130651800722*m^2*x^12+6957151150950*x^14+11*m^1
1*x^2+468765*m^9*x^4+1059893010*m^7*x^6+279691771260*m^5*x^8+9079996141062*m^3*x
^10+24133835554290*m*x^12+1551*m^10*x^2+16677375*m^8*x^4+15768085410*m^6*x^6+181
8135330660*m^4*x^8+22226933020446*m^2*x^10+11238474936150*x^12+m^11+96745*m^9*x^
2+380801190*m^7*x^4+158293212990*m^5*x^6+7587607623090*m^3*x^8+28336045738770*m*
x^10+143*m^10+3514005*m^8*x^2+5825106210*m^6*x^4+1059628145070*m^4*x^6+189307389
43710*m^2*x^8+13281834015450*x^10+9075*m^9+82295598*m^7*x^2+60431072570*m^5*x^4+
4558015784025*m^3*x^6+24503570194950*m*x^8+336765*m^8+1298935638*m^6*x^2+4204048
49150*m^4*x^4+11703493287585*m^2*x^6+11595251918250*x^8+8103018*m^7+14014513810*
m^5*x^2+1889780020755*m^3*x^4+15515657331075*m*x^6+132426294*m^6+102468500970*m^
4*x^2+5087634488145*m^2*x^4+7454090518875*x^6+1495875590*m^5+490955350391*m^3*x^
2+7041864340665*m*x^4+11641582810*m^4+1434440867211*m^2*x^2+3478575575475*x^4+60
936676581*m^3+2192684754645*m*x^2+203363952363*m^2+1159525191825*x^2+38718217093
5*m+316234143225)*x/(1+m)/(3+m)/(5+m)/(7+m)/(9+m)/(11+m)/(13+m)/(15+m)/(17+m)/(1
9+m)/(21+m)/(23+m)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 2*x^2 + 1)^5*(x^2 + 1)*(f*x)^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.293353, size = 1025, normalized size = 5.05 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 2*x^2 + 1)^5*(x^2 + 1)*(f*x)^m,x, algorithm="fricas")
[Out]
((m^11 + 121*m^10 + 6435*m^9 + 197835*m^8 + 3889578*m^7 + 51069018*m^6 + 4537144
70*m^5 + 2702025590*m^4 + 10431670821*m^3 + 24372200061*m^2 + 29985521895*m + 13
749310575)*x^23 + 11*(m^11 + 123*m^10 + 6635*m^9 + 206505*m^8 + 4103178*m^7 + 54
362574*m^6 + 486687830*m^5 + 2917013970*m^4 + 11320966021*m^3 + 26560342503*m^2
+ 32778930735*m + 15058768725)*x^21 + 55*(m^11 + 125*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 + 32368407579*m^2 + 40283194455*m + 18602008425)*x^17 + 3
30*(m^11 + 129*m^10 + 7283*m^9 + 236595*m^8 + 4890858*m^7 + 67166442*m^6 + 62080
5254*m^5 + 3825379590*m^4 + 15197565541*m^3 + 36337145829*m^2 + 45488935863*m +
21082276215)*x^15 + 462*(m^11 + 131*m^10 + 7515*m^9 + 248145*m^8 + 5213898*m^7 +
72748638*m^6 + 682569590*m^5 + 4264053730*m^4 + 17145560901*m^3 + 41408337231*m
^2 + 52237739295*m + 24325703325)*x^13 + 462*(m^11 + 133*m^10 + 7755*m^9 + 26053
5*m^8 + 5573898*m^7 + 79216434*m^6 + 756921110*m^5 + 4811326190*m^4 + 1965367130
1*m^3 + 48110244633*m^2 + 61333432335*m + 28748558475)*x^11 + 330*(m^11 + 135*m^
10 + 8003*m^9 + 273813*m^8 + 5975466*m^7 + 86750118*m^6 + 847550822*m^5 + 550950
1002*m^4 + 22992750373*m^3 + 57365875587*m^2 + 74253243015*m + 35137127025)*x^9
+ 165*(m^11 + 137*m^10 + 8259*m^9 + 288027*m^8 + 6423594*m^7 + 95564154*m^6 + 95
9352806*m^5 + 6421988758*m^4 + 27624338085*m^3 + 70930262349*m^2 + 94034286855*m
+ 45176306175)*x^7 + 55*(m^11 + 139*m^10 + 8523*m^9 + 303225*m^8 + 6923658*m^7
+ 105911022*m^6 + 1098746774*m^5 + 7643724530*m^4 + 34359636741*m^3 + 9250244523
9*m^2 + 128033897103*m + 63246828645)*x^5 + 11*(m^11 + 141*m^10 + 8795*m^9 + 319
455*m^8 + 7481418*m^7 + 118085058*m^6 + 1274046710*m^5 + 9315318270*m^4 + 446323
04581*m^3 + 130403715201*m^2 + 199334977695*m + 105411381075)*x^3 + (m^11 + 143*
m^10 + 9075*m^9 + 336765*m^8 + 8103018*m^7 + 132426294*m^6 + 1495875590*m^5 + 11
641582810*m^4 + 60936676581*m^3 + 203363952363*m^2 + 387182170935*m + 3162341432
25)*x)*(f*x)^m/(m^12 + 144*m^11 + 9218*m^10 + 345840*m^9 + 8439783*m^8 + 1405293
12*m^7 + 1628301884*m^6 + 13137458400*m^5 + 72578259391*m^4 + 264300628944*m^3 +
590546123298*m^2 + 703416314160*m + 316234143225)
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x)**m*(x**2+1)*(x**4+2*x**2+1)**5,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.296308, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 2*x^2 + 1)^5*(x^2 + 1)*(f*x)^m,x, algorithm="giac")
[Out]
Done