3.65 \(\int (f x)^m (1+x^2) (1+2 x^2+x^4)^5 \, dx\)

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 = 0.07, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {28, 270} \[ \frac {11 (f x)^{m+3}}{f^3 (m+3)}+\frac {55 (f x)^{m+5}}{f^5 (m+5)}+\frac {165 (f x)^{m+7}}{f^7 (m+7)}+\frac {330 (f x)^{m+9}}{f^9 (m+9)}+\frac {462 (f x)^{m+11}}{f^{11} (m+11)}+\frac {462 (f x)^{m+13}}{f^{13} (m+13)}+\frac {330 (f x)^{m+15}}{f^{15} (m+15)}+\frac {165 (f x)^{m+17}}{f^{17} (m+17)}+\frac {55 (f x)^{m+19}}{f^{19} (m+19)}+\frac {11 (f x)^{m+21}}{f^{21} (m+21)}+\frac {(f x)^{m+23}}{f^{23} (m+23)}+\frac {(f x)^{m+1}}{f (m+1)} \]

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))

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int (f x)^m \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int (f x)^m \left (1+x^2\right )^{11} \, dx\\ &=\int \left ((f x)^m+\frac {11 (f x)^{2+m}}{f^2}+\frac {55 (f x)^{4+m}}{f^4}+\frac {165 (f x)^{6+m}}{f^6}+\frac {330 (f x)^{8+m}}{f^8}+\frac {462 (f x)^{10+m}}{f^{10}}+\frac {462 (f x)^{12+m}}{f^{12}}+\frac {330 (f x)^{14+m}}{f^{14}}+\frac {165 (f x)^{16+m}}{f^{16}}+\frac {55 (f x)^{18+m}}{f^{18}}+\frac {11 (f x)^{20+m}}{f^{20}}+\frac {(f x)^{22+m}}{f^{22}}\right ) \, 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)}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 122, normalized size = 0.60 \[ x \left (\frac {x^{22}}{m+23}+\frac {11 x^{20}}{m+21}+\frac {55 x^{18}}{m+19}+\frac {165 x^{16}}{m+17}+\frac {330 x^{14}}{m+15}+\frac {462 x^{12}}{m+13}+\frac {462 x^{10}}{m+11}+\frac {330 x^8}{m+9}+\frac {165 x^6}{m+7}+\frac {55 x^4}{m+5}+\frac {11 x^2}{m+3}+\frac {1}{m+1}\right ) (f x)^m \]

Antiderivative was successfully verified.

[In]

Integrate[(f*x)^m*(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]

[Out]

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^2
0)/(21 + m) + x^22/(23 + m))

________________________________________________________________________________________

fricas [B]  time = 0.83, size = 759, normalized size = 3.74 \[ \text {result too large to display} \]

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, algorithm="fricas")

[Out]

((m^11 + 121*m^10 + 6435*m^9 + 197835*m^8 + 3889578*m^7 + 51069018*m^6 + 453714470*m^5 + 2702025590*m^4 + 1043
1670821*m^3 + 24372200061*m^2 + 29985521895*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 + 11320966021*m^3 + 26560342503*m^2 + 32778930
735*m + 15058768725)*x^21 + 55*(m^11 + 125*m^10 + 6843*m^9 + 215823*m^8 + 4339146*m^7 + 58085538*m^6 + 5246766
62*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 + 330*(m^11 + 129*m^10 + 7283*m^9 + 236595*m^8 + 4890
858*m^7 + 67166442*m^6 + 620805254*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^1
0 + 7755*m^9 + 260535*m^8 + 5573898*m^7 + 79216434*m^6 + 756921110*m^5 + 4811326190*m^4 + 19653671301*m^3 + 48
110244633*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 + 5509501002*m^4 + 22992750373*m^3 + 57365875587*m^2 + 74253243015*m + 3513712
7025)*x^9 + 165*(m^11 + 137*m^10 + 8259*m^9 + 288027*m^8 + 6423594*m^7 + 95564154*m^6 + 959352806*m^5 + 642198
8758*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 + 319455*m^8 + 7481418*m^7 + 118085
058*m^6 + 1274046710*m^5 + 9315318270*m^4 + 44632304581*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 + 11641582810*
m^4 + 60936676581*m^3 + 203363952363*m^2 + 387182170935*m + 316234143225)*x)*(f*x)^m/(m^12 + 144*m^11 + 9218*m
^10 + 345840*m^9 + 8439783*m^8 + 140529312*m^7 + 1628301884*m^6 + 13137458400*m^5 + 72578259391*m^4 + 26430062
8944*m^3 + 590546123298*m^2 + 703416314160*m + 316234143225)

________________________________________________________________________________________

giac [B]  time = 0.62, size = 1848, normalized size = 9.10 \[ \text {result too large to display} \]

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, algorithm="giac")

[Out]

((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 + 68
75*(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 + 4
2570*(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^2
1 + 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 + 28857216410*(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 + 10282782510*(f*x)^m*m^6*x^17 + 174273100210*(f*x)^m*m^4*x^19 + 292163
767533*(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 + 161398
3140*(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 + 165646455975*(f*x)^m*x^21 + 330*(f*x)^m*m^11*x^9 + 3582810*(f*
x)^m*m^9*x^11 + 2408820876*(f*x)^m*m^7*x^13 + 204865733820*(f*x)^m*m^5*x^15 + 2251106854425*(f*x)^m*m^3*x^17 +
1988025402825*(f*x)^m*m*x^19 + 44550*(f*x)^m*m^10*x^9 + 120367170*(f*x)^m*m^8*x^11 + 33609870756*(f*x)^m*m^6*
x^13 + 1262375264700*(f*x)^m*m^4*x^15 + 5340787250535*(f*x)^m*m^2*x^17 + 915414625125*(f*x)^m*x^19 + 165*(f*x)
^m*m^11*x^7 + 2640990*(f*x)^m*m^9*x^9 + 2575140876*(f*x)^m*m^7*x^11 + 315347150580*(f*x)^m*m^5*x^13 + 50151966
28530*(f*x)^m*m^3*x^15 + 6646727085075*(f*x)^m*m*x^17 + 22605*(f*x)^m*m^10*x^7 + 90358290*(f*x)^m*m^8*x^9 + 36
597992508*(f*x)^m*m^6*x^11 + 1969992823260*(f*x)^m*m^4*x^13 + 11991258123570*(f*x)^m*m^2*x^15 + 3069331390125*
(f*x)^m*x^17 + 55*(f*x)^m*m^11*x^5 + 1362735*(f*x)^m*m^9*x^7 + 1971903780*(f*x)^m*m^7*x^9 + 349697552820*(f*x)
^m*m^5*x^11 + 7921249136262*(f*x)^m*m^3*x^13 + 15011348834790*(f*x)^m*m*x^15 + 7645*(f*x)^m*m^10*x^5 + 4752445
5*(f*x)^m*m^8*x^7 + 28627538940*(f*x)^m*m^6*x^9 + 2222832699780*(f*x)^m*m^4*x^11 + 19130651800722*(f*x)^m*m^2*
x^13 + 6957151150950*(f*x)^m*x^15 + 11*(f*x)^m*m^11*x^3 + 468765*(f*x)^m*m^9*x^5 + 1059893010*(f*x)^m*m^7*x^7
+ 279691771260*(f*x)^m*m^5*x^9 + 9079996141062*(f*x)^m*m^3*x^11 + 24133835554290*(f*x)^m*m*x^13 + 1551*(f*x)^m
*m^10*x^3 + 16677375*(f*x)^m*m^8*x^5 + 15768085410*(f*x)^m*m^6*x^7 + 1818135330660*(f*x)^m*m^4*x^9 + 222269330
20446*(f*x)^m*m^2*x^11 + 11238474936150*(f*x)^m*x^13 + (f*x)^m*m^11*x + 96745*(f*x)^m*m^9*x^3 + 380801190*(f*x
)^m*m^7*x^5 + 158293212990*(f*x)^m*m^5*x^7 + 7587607623090*(f*x)^m*m^3*x^9 + 28336045738770*(f*x)^m*m*x^11 + 1
43*(f*x)^m*m^10*x + 3514005*(f*x)^m*m^8*x^3 + 5825106210*(f*x)^m*m^6*x^5 + 1059628145070*(f*x)^m*m^4*x^7 + 189
30738943710*(f*x)^m*m^2*x^9 + 13281834015450*(f*x)^m*x^11 + 9075*(f*x)^m*m^9*x + 82295598*(f*x)^m*m^7*x^3 + 60
431072570*(f*x)^m*m^5*x^5 + 4558015784025*(f*x)^m*m^3*x^7 + 24503570194950*(f*x)^m*m*x^9 + 336765*(f*x)^m*m^8*
x + 1298935638*(f*x)^m*m^6*x^3 + 420404849150*(f*x)^m*m^4*x^5 + 11703493287585*(f*x)^m*m^2*x^7 + 1159525191825
0*(f*x)^m*x^9 + 8103018*(f*x)^m*m^7*x + 14014513810*(f*x)^m*m^5*x^3 + 1889780020755*(f*x)^m*m^3*x^5 + 15515657
331075*(f*x)^m*m*x^7 + 132426294*(f*x)^m*m^6*x + 102468500970*(f*x)^m*m^4*x^3 + 5087634488145*(f*x)^m*m^2*x^5
+ 7454090518875*(f*x)^m*x^7 + 1495875590*(f*x)^m*m^5*x + 490955350391*(f*x)^m*m^3*x^3 + 7041864340665*(f*x)^m*
m*x^5 + 11641582810*(f*x)^m*m^4*x + 1434440867211*(f*x)^m*m^2*x^3 + 3478575575475*(f*x)^m*x^5 + 60936676581*(f
*x)^m*m^3*x + 2192684754645*(f*x)^m*m*x^3 + 203363952363*(f*x)^m*m^2*x + 1159525191825*(f*x)^m*x^3 + 387182170
935*(f*x)^m*m*x + 316234143225*(f*x)^m*x)/(m^12 + 144*m^11 + 9218*m^10 + 345840*m^9 + 8439783*m^8 + 140529312*
m^7 + 1628301884*m^6 + 13137458400*m^5 + 72578259391*m^4 + 264300628944*m^3 + 590546123298*m^2 + 703416314160*
m + 316234143225)

________________________________________________________________________________________

maple [B]  time = 0.01, size = 1121, normalized size = 5.52 \[ \text {result too large to display} \]

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+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+4513
4958*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+3726310
5*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+759091
410*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+1
0282782510*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^1
2+1613983140*m^7*x^14+93862508190*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+165646455975*x^20+330*m^11*x^8+358
2810*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+915414625125*x^18+1
65*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+11991258123570*m^2*x^14+306933
1390125*x^16+55*m^11*x^4+1362735*m^9*x^6+1971903780*m^7*x^8+349697552820*m^5*x^10+7921249136262*m^3*x^12+15011
348834790*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^11*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+1818135330660*m^4*x^8+22226933020
446*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+18930738943710*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+420404849150*m^4*x^4+11703493287585*m^2*x^6+11595251918250*x^8+8103018*m^7+14
014513810*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+143444086721
1*m^2*x^2+3478575575475*x^4+60936676581*m^3+2192684754645*m*x^2+203363952363*m^2+1159525191825*x^2+38718217093
5*m+316234143225)*x/(m+1)/(m+3)/(m+5)/(m+7)/(m+9)/(m+11)/(m+13)/(m+15)/(m+17)/(m+19)/(m+21)/(m+23)

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maxima [A]  time = 0.97, size = 192, normalized size = 0.95 \[ \frac {f^{m} x^{23} x^{m}}{m + 23} + \frac {11 \, f^{m} x^{21} x^{m}}{m + 21} + \frac {55 \, f^{m} x^{19} x^{m}}{m + 19} + \frac {165 \, f^{m} x^{17} x^{m}}{m + 17} + \frac {330 \, f^{m} x^{15} x^{m}}{m + 15} + \frac {462 \, f^{m} x^{13} x^{m}}{m + 13} + \frac {462 \, f^{m} x^{11} x^{m}}{m + 11} + \frac {330 \, f^{m} x^{9} x^{m}}{m + 9} + \frac {165 \, f^{m} x^{7} x^{m}}{m + 7} + \frac {55 \, f^{m} x^{5} x^{m}}{m + 5} + \frac {11 \, f^{m} x^{3} x^{m}}{m + 3} + \frac {\left (f x\right )^{m + 1}}{f {\left (m + 1\right )}} \]

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, algorithm="maxima")

[Out]

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))

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mupad [B]  time = 1.25, size = 1483, normalized size = 7.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 1)*(f*x)^m*(2*x^2 + x^4 + 1)^5,x)

[Out]

(x^3*(f*x)^m*(2192684754645*m + 1434440867211*m^2 + 490955350391*m^3 + 102468500970*m^4 + 14014513810*m^5 + 12
98935638*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 + 8
439783*m^8 + 345840*m^9 + 9218*m^10 + 144*m^11 + m^12 + 316234143225) + (x^19*(f*x)^m*(1988025402825*m + 16048
42704135*m^2 + 680615362515*m^3 + 174273100210*m^4 + 28857216410*m^5 + 3194704590*m^6 + 238653030*m^7 + 118702
65*m^8 + 376365*m^9 + 6875*m^10 + 55*m^11 + 915414625125))/(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
+ 144*m^11 + m^12 + 316234143225) + (x^11*(f*x)^m*(28336045738770*m + 22226933020446*m^2 + 9079996141062*m^3
+ 2222832699780*m^4 + 349697552820*m^5 + 36597992508*m^6 + 2575140876*m^7 + 120367170*m^8 + 3582810*m^9 + 6144
6*m^10 + 462*m^11 + 13281834015450))/(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 + 144*m^11 + m^12 + 3
16234143225) + (x^21*(f*x)^m*(360568238085*m + 292163767533*m^2 + 124530626231*m^3 + 32087153670*m^4 + 5353566
130*m^5 + 597988314*m^6 + 45134958*m^7 + 2271555*m^8 + 72985*m^9 + 1353*m^10 + 11*m^11 + 165646455975))/(70341
6314160*m + 590546123298*m^2 + 264300628944*m^3 + 72578259391*m^4 + 13137458400*m^5 + 1628301884*m^6 + 1405293
12*m^7 + 8439783*m^8 + 345840*m^9 + 9218*m^10 + 144*m^11 + m^12 + 316234143225) + (x^5*(f*x)^m*(7041864340665*
m + 5087634488145*m^2 + 1889780020755*m^3 + 420404849150*m^4 + 60431072570*m^5 + 5825106210*m^6 + 380801190*m^
7 + 16677375*m^8 + 468765*m^9 + 7645*m^10 + 55*m^11 + 3478575575475))/(703416314160*m + 590546123298*m^2 + 264
300628944*m^3 + 72578259391*m^4 + 13137458400*m^5 + 1628301884*m^6 + 140529312*m^7 + 8439783*m^8 + 345840*m^9
+ 9218*m^10 + 144*m^11 + m^12 + 316234143225) + (x^17*(f*x)^m*(6646727085075*m + 5340787250535*m^2 + 225110685
4425*m^3 + 572017996770*m^4 + 93862508190*m^5 + 10282782510*m^6 + 759091410*m^7 + 37263105*m^8 + 1164735*m^9 +
20955*m^10 + 165*m^11 + 3069331390125))/(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 + 144*m^11 + m^12
+ 316234143225) + (x*(f*x)^m*(387182170935*m + 203363952363*m^2 + 60936676581*m^3 + 11641582810*m^4 + 1495875
590*m^5 + 132426294*m^6 + 8103018*m^7 + 336765*m^8 + 9075*m^9 + 143*m^10 + m^11 + 316234143225))/(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 + 144*m^11 + m^12 + 316234143225) + (x^23*(f*x)^m*(29985521895*m + 2437
2200061*m^2 + 10431670821*m^3 + 2702025590*m^4 + 453714470*m^5 + 51069018*m^6 + 3889578*m^7 + 197835*m^8 + 643
5*m^9 + 121*m^10 + m^11 + 13749310575))/(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 + 144*m^11 + m^12
+ 316234143225) + (x^7*(f*x)^m*(15515657331075*m + 11703493287585*m^2 + 4558015784025*m^3 + 1059628145070*m^4
+ 158293212990*m^5 + 15768085410*m^6 + 1059893010*m^7 + 47524455*m^8 + 1362735*m^9 + 22605*m^10 + 165*m^11 + 7
454090518875))/(703416314160*m + 590546123298*m^2 + 264300628944*m^3 + 72578259391*m^4 + 13137458400*m^5 + 162
8301884*m^6 + 140529312*m^7 + 8439783*m^8 + 345840*m^9 + 9218*m^10 + 144*m^11 + m^12 + 316234143225) + (x^15*(
f*x)^m*(15011348834790*m + 11991258123570*m^2 + 5015196628530*m^3 + 1262375264700*m^4 + 204865733820*m^5 + 221
64925860*m^6 + 1613983140*m^7 + 78076350*m^8 + 2403390*m^9 + 42570*m^10 + 330*m^11 + 6957151150950))/(70341631
4160*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 + 144*m^11 + m^12 + 316234143225) + (x^9*(f*x)^m*(24503570194950*m
+ 18930738943710*m^2 + 7587607623090*m^3 + 1818135330660*m^4 + 279691771260*m^5 + 28627538940*m^6 + 1971903780
*m^7 + 90358290*m^8 + 2640990*m^9 + 44550*m^10 + 330*m^11 + 11595251918250))/(703416314160*m + 590546123298*m^
2 + 264300628944*m^3 + 72578259391*m^4 + 13137458400*m^5 + 1628301884*m^6 + 140529312*m^7 + 8439783*m^8 + 3458
40*m^9 + 9218*m^10 + 144*m^11 + m^12 + 316234143225) + (x^13*(f*x)^m*(24133835554290*m + 19130651800722*m^2 +
7921249136262*m^3 + 1969992823260*m^4 + 315347150580*m^5 + 33609870756*m^6 + 2408820876*m^7 + 114642990*m^8 +
3471930*m^9 + 60522*m^10 + 462*m^11 + 11238474936150))/(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 + 1
44*m^11 + m^12 + 316234143225)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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