3.1.55 \(\int (f x)^m (d+e x^2) (1+2 x^2+x^4)^5 \, dx\) [55]
Optimal. Leaf size=269 \[ \frac {d (f x)^{1+m}}{f (1+m)}+\frac {(10 d+e) (f x)^{3+m}}{f^3 (3+m)}+\frac {5 (9 d+2 e) (f x)^{5+m}}{f^5 (5+m)}+\frac {15 (8 d+3 e) (f x)^{7+m}}{f^7 (7+m)}+\frac {30 (7 d+4 e) (f x)^{9+m}}{f^9 (9+m)}+\frac {42 (6 d+5 e) (f x)^{11+m}}{f^{11} (11+m)}+\frac {42 (5 d+6 e) (f x)^{13+m}}{f^{13} (13+m)}+\frac {30 (4 d+7 e) (f x)^{15+m}}{f^{15} (15+m)}+\frac {15 (3 d+8 e) (f x)^{17+m}}{f^{17} (17+m)}+\frac {5 (2 d+9 e) (f x)^{19+m}}{f^{19} (19+m)}+\frac {(d+10 e) (f x)^{21+m}}{f^{21} (21+m)}+\frac {e (f x)^{23+m}}{f^{23} (23+m)} \]
[Out]
d*(f*x)^(1+m)/f/(1+m)+(10*d+e)*(f*x)^(3+m)/f^3/(3+m)+5*(9*d+2*e)*(f*x)^(5+m)/f^5/(5+m)+15*(8*d+3*e)*(f*x)^(7+m
)/f^7/(7+m)+30*(7*d+4*e)*(f*x)^(9+m)/f^9/(9+m)+42*(6*d+5*e)*(f*x)^(11+m)/f^11/(11+m)+42*(5*d+6*e)*(f*x)^(13+m)
/f^13/(13+m)+30*(4*d+7*e)*(f*x)^(15+m)/f^15/(15+m)+15*(3*d+8*e)*(f*x)^(17+m)/f^17/(17+m)+5*(2*d+9*e)*(f*x)^(19
+m)/f^19/(19+m)+(d+10*e)*(f*x)^(21+m)/f^21/(21+m)+e*(f*x)^(23+m)/f^23/(23+m)
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {28, 459}
\begin {gather*} \frac {(d+10 e) (f x)^{m+21}}{f^{21} (m+21)}+\frac {5 (2 d+9 e) (f x)^{m+19}}{f^{19} (m+19)}+\frac {15 (3 d+8 e) (f x)^{m+17}}{f^{17} (m+17)}+\frac {30 (4 d+7 e) (f x)^{m+15}}{f^{15} (m+15)}+\frac {42 (5 d+6 e) (f x)^{m+13}}{f^{13} (m+13)}+\frac {42 (6 d+5 e) (f x)^{m+11}}{f^{11} (m+11)}+\frac {30 (7 d+4 e) (f x)^{m+9}}{f^9 (m+9)}+\frac {15 (8 d+3 e) (f x)^{m+7}}{f^7 (m+7)}+\frac {5 (9 d+2 e) (f x)^{m+5}}{f^5 (m+5)}+\frac {(10 d+e) (f x)^{m+3}}{f^3 (m+3)}+\frac {d (f x)^{m+1}}{f (m+1)}+\frac {e (f x)^{m+23}}{f^{23} (m+23)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(f*x)^m*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]
[Out]
(d*(f*x)^(1 + m))/(f*(1 + m)) + ((10*d + e)*(f*x)^(3 + m))/(f^3*(3 + m)) + (5*(9*d + 2*e)*(f*x)^(5 + m))/(f^5*
(5 + m)) + (15*(8*d + 3*e)*(f*x)^(7 + m))/(f^7*(7 + m)) + (30*(7*d + 4*e)*(f*x)^(9 + m))/(f^9*(9 + m)) + (42*(
6*d + 5*e)*(f*x)^(11 + m))/(f^11*(11 + m)) + (42*(5*d + 6*e)*(f*x)^(13 + m))/(f^13*(13 + m)) + (30*(4*d + 7*e)
*(f*x)^(15 + m))/(f^15*(15 + m)) + (15*(3*d + 8*e)*(f*x)^(17 + m))/(f^17*(17 + m)) + (5*(2*d + 9*e)*(f*x)^(19
+ m))/(f^19*(19 + m)) + ((d + 10*e)*(f*x)^(21 + m))/(f^21*(21 + m)) + (e*(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 459
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int (f x)^m \left (1+x^2\right )^{10} \left (d+e x^2\right ) \, dx\\ &=\int \left (d (f x)^m+\frac {(10 d+e) (f x)^{2+m}}{f^2}+\frac {5 (9 d+2 e) (f x)^{4+m}}{f^4}+\frac {15 (8 d+3 e) (f x)^{6+m}}{f^6}+\frac {30 (7 d+4 e) (f x)^{8+m}}{f^8}+\frac {42 (6 d+5 e) (f x)^{10+m}}{f^{10}}+\frac {42 (5 d+6 e) (f x)^{12+m}}{f^{12}}+\frac {30 (4 d+7 e) (f x)^{14+m}}{f^{14}}+\frac {15 (3 d+8 e) (f x)^{16+m}}{f^{16}}+\frac {5 (2 d+9 e) (f x)^{18+m}}{f^{18}}+\frac {(d+10 e) (f x)^{20+m}}{f^{20}}+\frac {e (f x)^{22+m}}{f^{22}}\right ) \, dx\\ &=\frac {d (f x)^{1+m}}{f (1+m)}+\frac {(10 d+e) (f x)^{3+m}}{f^3 (3+m)}+\frac {5 (9 d+2 e) (f x)^{5+m}}{f^5 (5+m)}+\frac {15 (8 d+3 e) (f x)^{7+m}}{f^7 (7+m)}+\frac {30 (7 d+4 e) (f x)^{9+m}}{f^9 (9+m)}+\frac {42 (6 d+5 e) (f x)^{11+m}}{f^{11} (11+m)}+\frac {42 (5 d+6 e) (f x)^{13+m}}{f^{13} (13+m)}+\frac {30 (4 d+7 e) (f x)^{15+m}}{f^{15} (15+m)}+\frac {15 (3 d+8 e) (f x)^{17+m}}{f^{17} (17+m)}+\frac {5 (2 d+9 e) (f x)^{19+m}}{f^{19} (19+m)}+\frac {(d+10 e) (f x)^{21+m}}{f^{21} (21+m)}+\frac {e (f x)^{23+m}}{f^{23} (23+m)}\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 189, normalized size = 0.70 \begin {gather*} x (f x)^m \left (\frac {d}{1+m}+\frac {(10 d+e) x^2}{3+m}+\frac {5 (9 d+2 e) x^4}{5+m}+\frac {15 (8 d+3 e) x^6}{7+m}+\frac {30 (7 d+4 e) x^8}{9+m}+\frac {42 (6 d+5 e) x^{10}}{11+m}+\frac {42 (5 d+6 e) x^{12}}{13+m}+\frac {30 (4 d+7 e) x^{14}}{15+m}+\frac {15 (3 d+8 e) x^{16}}{17+m}+\frac {5 (2 d+9 e) x^{18}}{19+m}+\frac {(d+10 e) x^{20}}{21+m}+\frac {e x^{22}}{23+m}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(f*x)^m*(d + e*x^2)*(1 + 2*x^2 + x^4)^5,x]
[Out]
x*(f*x)^m*(d/(1 + m) + ((10*d + e)*x^2)/(3 + m) + (5*(9*d + 2*e)*x^4)/(5 + m) + (15*(8*d + 3*e)*x^6)/(7 + m) +
(30*(7*d + 4*e)*x^8)/(9 + m) + (42*(6*d + 5*e)*x^10)/(11 + m) + (42*(5*d + 6*e)*x^12)/(13 + m) + (30*(4*d + 7
*e)*x^14)/(15 + m) + (15*(3*d + 8*e)*x^16)/(17 + m) + (5*(2*d + 9*e)*x^18)/(19 + m) + ((d + 10*e)*x^20)/(21 +
m) + (e*x^22)/(23 + m))
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2294\) vs.
\(2(269)=538\).
time = 0.04, size = 2295, normalized size = 8.53
| | |
| method |
result |
size |
| | |
| gosper |
\(\text {Expression too large to display}\) |
\(2295\) |
| risch |
\(\text {Expression too large to display}\) |
\(2295\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x)^m*(e*x^2+d)*(x^4+2*x^2+1)^5,x,method=_RETURNVERBOSE)
[Out]
(f*x)^m*(e*m^11*x^22+121*e*m^10*x^22+d*m^11*x^20+10*e*m^11*x^20+6435*e*m^9*x^22+123*d*m^10*x^20+1230*e*m^10*x^
20+197835*e*m^8*x^22+10*d*m^11*x^18+6635*d*m^9*x^20+45*e*m^11*x^18+66350*e*m^9*x^20+3889578*e*m^7*x^22+1250*d*
m^10*x^18+206505*d*m^8*x^20+5625*e*m^10*x^18+2065050*e*m^8*x^20+51069018*e*m^6*x^22+45*d*m^11*x^16+68430*d*m^9
*x^18+4103178*d*m^7*x^20+120*e*m^11*x^16+307935*e*m^9*x^18+41031780*e*m^7*x^20+453714470*e*m^5*x^22+5715*d*m^1
0*x^16+2158230*d*m^8*x^18+54362574*d*m^6*x^20+15240*e*m^10*x^16+9712035*e*m^8*x^18+543625740*e*m^6*x^20+270202
5590*e*m^4*x^22+120*d*m^11*x^14+317655*d*m^9*x^16+43391460*d*m^7*x^18+486687830*d*m^5*x^20+210*e*m^11*x^14+847
080*e*m^9*x^16+195261570*e*m^7*x^18+4866878300*e*m^5*x^20+10431670821*e*m^3*x^22+15480*d*m^10*x^14+10162665*d*
m^8*x^16+580855380*d*m^6*x^18+2917013970*d*m^4*x^20+27090*e*m^10*x^14+27100440*e*m^8*x^16+2613849210*e*m^6*x^1
8+29170139700*e*m^4*x^20+24372200061*e*m^2*x^22+210*d*m^11*x^12+873960*d*m^9*x^14+207024930*d*m^7*x^16+5246766
620*d*m^5*x^18+11320966021*d*m^3*x^20+252*e*m^11*x^12+1529430*e*m^9*x^14+552066480*e*m^7*x^16+23610449790*e*m^
5*x^18+113209660210*e*m^3*x^20+29985521895*e*m*x^22+27510*d*m^10*x^12+28391400*d*m^8*x^14+2804395230*d*m^6*x^1
6+31686018220*d*m^4*x^18+26560342503*d*m^2*x^20+33012*e*m^10*x^12+49684950*e*m^8*x^14+7478387280*e*m^6*x^16+14
2587081990*e*m^4*x^18+265603425030*e*m^2*x^20+13749310575*e*x^22+252*d*m^11*x^10+1578150*d*m^9*x^12+586902960*
d*m^7*x^14+25598865870*d*m^5*x^16+123748247730*d*m^3*x^18+32778930735*d*m*x^20+210*e*m^11*x^10+1893780*e*m^9*x
^12+1027080180*e*m^7*x^14+68263642320*e*m^5*x^16+556867114785*e*m^3*x^18+327789307350*e*m*x^20+33516*d*m^10*x^
10+52110450*d*m^8*x^12+8059973040*d*m^6*x^14+156004908210*d*m^4*x^16+291789582570*d*m^2*x^18+15058768725*d*x^2
0+27930*e*m^10*x^10+62532540*e*m^8*x^12+14104952820*e*m^6*x^14+416013088560*e*m^4*x^16+1313053121565*e*m^2*x^1
8+150587687250*e*x^20+210*d*m^11*x^8+1954260*d*m^9*x^10+1094918580*d*m^7*x^12+74496630480*d*m^5*x^14+613938233
025*d*m^3*x^16+361459164150*d*m*x^18+120*e*m^11*x^8+1628550*e*m^9*x^10+1313902296*e*m^7*x^12+130369103340*e*m^
5*x^14+1637168621400*e*m^3*x^16+1626566238675*e*m*x^18+28350*d*m^10*x^8+65654820*d*m^8*x^10+15277213980*d*m^6*
x^12+459045550800*d*m^4*x^14+1456578341055*d*m^2*x^16+166439022750*d*x^18+16200*e*m^10*x^8+54712350*e*m^8*x^10
+18332656776*e*m^6*x^12+803329713900*e*m^4*x^14+3884208909480*e*m^2*x^16+748975602375*e*x^18+120*d*m^11*x^6+16
80630*d*m^9*x^8+1404622296*d*m^7*x^10+143339613900*d*m^5*x^12+1823707864920*d*m^3*x^14+1812743750475*d*m*x^16+
45*e*m^11*x^6+960360*e*m^9*x^8+1170518580*e*m^7*x^10+172007536680*e*m^5*x^12+3191488763610*e*m^3*x^14+48339833
34600*e*m*x^16+16440*d*m^10*x^6+57500730*d*m^8*x^8+19962541368*d*m^6*x^10+895451283300*d*m^4*x^12+436045749948
0*d*m^2*x^14+837090379125*d*x^16+6165*e*m^10*x^6+32857560*e*m^8*x^8+16635451140*e*m^6*x^10+1074541539960*e*m^4
*x^12+7630800624090*e*m^2*x^14+2232241011000*e*x^16+45*d*m^11*x^4+991080*d*m^9*x^6+1254847860*d*m^7*x^8+190744
119720*d*m^5*x^10+3600567789210*d*m^3*x^12+5458672303560*d*m*x^14+10*e*m^11*x^4+371655*e*m^9*x^6+717055920*e*m
^7*x^8+158953433100*e*m^5*x^10+4320681347052*e*m^3*x^12+9552676531230*e*m*x^14+6255*d*m^10*x^4+34563240*d*m^8*
x^6+18217524780*d*m^6*x^8+1212454199880*d*m^4*x^10+8695750818510*d*m^2*x^12+2529873145800*d*x^14+1390*e*m^10*x
^4+12961215*e*m^8*x^6+10410014160*e*m^6*x^8+1010378499900*e*m^4*x^10+10434900982212*e*m^2*x^12+4427278005150*e
*x^14+10*d*m^11*x^2+383535*d*m^9*x^4+770831280*d*m^7*x^6+177985672620*d*m^5*x^8+4952725167852*d*m^3*x^10+10969
925251950*d*m*x^12+e*m^11*x^2+85230*e*m^9*x^4+289061730*e*m^7*x^6+101706098640*e*m^5*x^8+4127270973210*e*m^3*x
^10+13163910302340*e*m*x^12+1410*d*m^10*x^2+13645125*d*m^8*x^4+11467698480*d*m^6*x^6+1156995210420*d*m^4*x^8+1
2123781647516*d*m^2*x^10+5108397698250*d*x^12+141*e*m^10*x^2+3032250*e*m^8*x^4+4300386930*e*m^6*x^6+6611401202
40*e*m^4*x^8+10103151372930*e*m^2*x^10+6130077237900*e*x^12+d*m^11+87950*d*m^9*x^2+311564610*d*m^7*x^4+1151223
36720*d*m^5*x^6+4828477578330*d*m^3*x^8+15456024948420*d*m*x^10+8795*e*m^9*x^2+69236580*e*m^7*x^4+43170876270*
e*m^5*x^6+2759130044760*e*m^3*x^8+12880020790350*e*m*x^10+143*d*m^10+3194550*d*m^8*x^2+4765995990*d*m^6*x^4+77
0638650960*d*m^4*x^6+12046833873270*d*m^2*x^8+7244636735700*d*x^10+319455*e*m^8*x^2+1059110220*e*m^6*x^4+28898
9494110*e*m^4*x^6+6883905070440*e*m^2*x^8+6037197279750*e*x^10+9075*d*m^9+74814180*d*m^7*x^2+49443604830*d*m^5
*x^4+3314920570200*d*m^3*x^6+15593181033150*d*m*x^8+7481418*e*m^7*x^2+10987467740*e*m^5*x^4+1243095213825*e*m^
3*x^6+8910389161800*e*m*x^8+336765*d*m^8+1180850580*d*m^6*x^2+343967603850*d*m^4*x^4+8511631481880*d*m^2*x^6+7
378796675250*d*x^8+118085058*e*m^6*x^2+76437245300*e*m^4*x^4+3191861805705*e*m^2*x^6+4216455243000*e*x^8+81030
18*d*m^7+12740467100*d*m^5*x^2+1546183653345*d*m^3*x^4+11284114422600*d*m*x^6+1274046710*e*m^5*x^2+34359636741
0*e*m^3*x^4+4231542908475*e*m*x^6+132426294*d*m^6+93153182700*d*m^4*x^2+4162610035755*d*m^2*x^4+5421156741000*
d*x^6+9315318270*e*m^4*x^2+925024452390*e*m^2*x...
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Maxima [A]
time = 0.34, size = 405, normalized size = 1.51 \begin {gather*} \frac {f^{m} x^{23} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 23} + \frac {d f^{m} x^{21} x^{m}}{m + 21} + \frac {10 \, f^{m} x^{21} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 21} + \frac {10 \, d f^{m} x^{19} x^{m}}{m + 19} + \frac {45 \, f^{m} x^{19} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 19} + \frac {45 \, d f^{m} x^{17} x^{m}}{m + 17} + \frac {120 \, f^{m} x^{17} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 17} + \frac {120 \, d f^{m} x^{15} x^{m}}{m + 15} + \frac {210 \, f^{m} x^{15} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 15} + \frac {210 \, d f^{m} x^{13} x^{m}}{m + 13} + \frac {252 \, f^{m} x^{13} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 13} + \frac {252 \, d f^{m} x^{11} x^{m}}{m + 11} + \frac {210 \, f^{m} x^{11} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 11} + \frac {210 \, d f^{m} x^{9} x^{m}}{m + 9} + \frac {120 \, f^{m} x^{9} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 9} + \frac {120 \, d f^{m} x^{7} x^{m}}{m + 7} + \frac {45 \, f^{m} x^{7} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 7} + \frac {45 \, d f^{m} x^{5} x^{m}}{m + 5} + \frac {10 \, f^{m} x^{5} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 5} + \frac {10 \, d f^{m} x^{3} x^{m}}{m + 3} + \frac {f^{m} x^{3} e^{\left (m \log \left (x\right ) + 1\right )}}{m + 3} + \frac {\left (f x\right )^{m + 1} d}{f {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x)^m*(e*x^2+d)*(x^4+2*x^2+1)^5,x, algorithm="maxima")
[Out]
f^m*x^23*e^(m*log(x) + 1)/(m + 23) + d*f^m*x^21*x^m/(m + 21) + 10*f^m*x^21*e^(m*log(x) + 1)/(m + 21) + 10*d*f^
m*x^19*x^m/(m + 19) + 45*f^m*x^19*e^(m*log(x) + 1)/(m + 19) + 45*d*f^m*x^17*x^m/(m + 17) + 120*f^m*x^17*e^(m*l
og(x) + 1)/(m + 17) + 120*d*f^m*x^15*x^m/(m + 15) + 210*f^m*x^15*e^(m*log(x) + 1)/(m + 15) + 210*d*f^m*x^13*x^
m/(m + 13) + 252*f^m*x^13*e^(m*log(x) + 1)/(m + 13) + 252*d*f^m*x^11*x^m/(m + 11) + 210*f^m*x^11*e^(m*log(x) +
1)/(m + 11) + 210*d*f^m*x^9*x^m/(m + 9) + 120*f^m*x^9*e^(m*log(x) + 1)/(m + 9) + 120*d*f^m*x^7*x^m/(m + 7) +
45*f^m*x^7*e^(m*log(x) + 1)/(m + 7) + 45*d*f^m*x^5*x^m/(m + 5) + 10*f^m*x^5*e^(m*log(x) + 1)/(m + 5) + 10*d*f^
m*x^3*x^m/(m + 3) + f^m*x^3*e^(m*log(x) + 1)/(m + 3) + (f*x)^(m + 1)*d/(f*(m + 1))
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1495 vs.
\(2 (280) = 560\).
time = 0.38, size = 1495, normalized size = 5.56 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x)^m*(e*x^2+d)*(x^4+2*x^2+1)^5,x, algorithm="fricas")
[Out]
((d*m^11 + 123*d*m^10 + 6635*d*m^9 + 206505*d*m^8 + 4103178*d*m^7 + 54362574*d*m^6 + 486687830*d*m^5 + 2917013
970*d*m^4 + 11320966021*d*m^3 + 26560342503*d*m^2 + 32778930735*d*m + 15058768725*d)*x^21 + 10*(d*m^11 + 125*d
*m^10 + 6843*d*m^9 + 215823*d*m^8 + 4339146*d*m^7 + 58085538*d*m^6 + 524676662*d*m^5 + 3168601822*d*m^4 + 1237
4824773*d*m^3 + 29178958257*d*m^2 + 36145916415*d*m + 16643902275*d)*x^19 + 45*(d*m^11 + 127*d*m^10 + 7059*d*m
^9 + 225837*d*m^8 + 4600554*d*m^7 + 62319894*d*m^6 + 568863686*d*m^5 + 3466775738*d*m^4 + 13643071845*d*m^3 +
32368407579*d*m^2 + 40283194455*d*m + 18602008425*d)*x^17 + 120*(d*m^11 + 129*d*m^10 + 7283*d*m^9 + 236595*d*m
^8 + 4890858*d*m^7 + 67166442*d*m^6 + 620805254*d*m^5 + 3825379590*d*m^4 + 15197565541*d*m^3 + 36337145829*d*m
^2 + 45488935863*d*m + 21082276215*d)*x^15 + 210*(d*m^11 + 131*d*m^10 + 7515*d*m^9 + 248145*d*m^8 + 5213898*d*
m^7 + 72748638*d*m^6 + 682569590*d*m^5 + 4264053730*d*m^4 + 17145560901*d*m^3 + 41408337231*d*m^2 + 5223773929
5*d*m + 24325703325*d)*x^13 + 252*(d*m^11 + 133*d*m^10 + 7755*d*m^9 + 260535*d*m^8 + 5573898*d*m^7 + 79216434*
d*m^6 + 756921110*d*m^5 + 4811326190*d*m^4 + 19653671301*d*m^3 + 48110244633*d*m^2 + 61333432335*d*m + 2874855
8475*d)*x^11 + 210*(d*m^11 + 135*d*m^10 + 8003*d*m^9 + 273813*d*m^8 + 5975466*d*m^7 + 86750118*d*m^6 + 8475508
22*d*m^5 + 5509501002*d*m^4 + 22992750373*d*m^3 + 57365875587*d*m^2 + 74253243015*d*m + 35137127025*d)*x^9 + 1
20*(d*m^11 + 137*d*m^10 + 8259*d*m^9 + 288027*d*m^8 + 6423594*d*m^7 + 95564154*d*m^6 + 959352806*d*m^5 + 64219
88758*d*m^4 + 27624338085*d*m^3 + 70930262349*d*m^2 + 94034286855*d*m + 45176306175*d)*x^7 + 45*(d*m^11 + 139*
d*m^10 + 8523*d*m^9 + 303225*d*m^8 + 6923658*d*m^7 + 105911022*d*m^6 + 1098746774*d*m^5 + 7643724530*d*m^4 + 3
4359636741*d*m^3 + 92502445239*d*m^2 + 128033897103*d*m + 63246828645*d)*x^5 + 10*(d*m^11 + 141*d*m^10 + 8795*
d*m^9 + 319455*d*m^8 + 7481418*d*m^7 + 118085058*d*m^6 + 1274046710*d*m^5 + 9315318270*d*m^4 + 44632304581*d*m
^3 + 130403715201*d*m^2 + 199334977695*d*m + 105411381075*d)*x^3 + (d*m^11 + 143*d*m^10 + 9075*d*m^9 + 336765*
d*m^8 + 8103018*d*m^7 + 132426294*d*m^6 + 1495875590*d*m^5 + 11641582810*d*m^4 + 60936676581*d*m^3 + 203363952
363*d*m^2 + 387182170935*d*m + 316234143225*d)*x + ((m^11 + 121*m^10 + 6435*m^9 + 197835*m^8 + 3889578*m^7 + 5
1069018*m^6 + 453714470*m^5 + 2702025590*m^4 + 10431670821*m^3 + 24372200061*m^2 + 29985521895*m + 13749310575
)*x^23 + 10*(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 + 32778930735*m + 15058768725)*x^21 + 45*(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 + 120*(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 +
210*(m^11 + 129*m^10 + 7283*m^9 + 236595*m^8 + 4890858*m^7 + 67166442*m^6 + 620805254*m^5 + 3825379590*m^4 + 1
5197565541*m^3 + 36337145829*m^2 + 45488935863*m + 21082276215)*x^15 + 252*(m^11 + 131*m^10 + 7515*m^9 + 24814
5*m^8 + 5213898*m^7 + 72748638*m^6 + 682569590*m^5 + 4264053730*m^4 + 17145560901*m^3 + 41408337231*m^2 + 5223
7739295*m + 24325703325)*x^13 + 210*(m^11 + 133*m^10 + 7755*m^9 + 260535*m^8 + 5573898*m^7 + 79216434*m^6 + 75
6921110*m^5 + 4811326190*m^4 + 19653671301*m^3 + 48110244633*m^2 + 61333432335*m + 28748558475)*x^11 + 120*(m^
11 + 135*m^10 + 8003*m^9 + 273813*m^8 + 5975466*m^7 + 86750118*m^6 + 847550822*m^5 + 5509501002*m^4 + 22992750
373*m^3 + 57365875587*m^2 + 74253243015*m + 35137127025)*x^9 + 45*(m^11 + 137*m^10 + 8259*m^9 + 288027*m^8 + 6
423594*m^7 + 95564154*m^6 + 959352806*m^5 + 6421988758*m^4 + 27624338085*m^3 + 70930262349*m^2 + 94034286855*m
+ 45176306175)*x^7 + 10*(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 + 92502445239*m^2 + 128033897103*m + 63246828645)*x^5 + (m^11 + 141*m^10
+ 8795*m^9 + 319455*m^8 + 7481418*m^7 + 118085058*m^6 + 1274046710*m^5 + 9315318270*m^4 + 44632304581*m^3 + 1
30403715201*m^2 + 199334977695*m + 105411381075)*x^3)*e)*(f*x)^m/(m^12 + 144*m^11 + 9218*m^10 + 345840*m^9 + 8
439783*m^8 + 140529312*m^7 + 1628301884*m^6 + 13137458400*m^5 + 72578259391*m^4 + 264300628944*m^3 + 590546123
298*m^2 + 703416314160*m + 316234143225)
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 21612 vs.
\(2 (228) = 456\).
time = 3.44, size = 21612, normalized size = 80.34 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x)**m*(e*x**2+d)*(x**4+2*x**2+1)**5,x)
[Out]
Piecewise(((-d/(2*x**2) - 5*d/(2*x**4) - 15*d/(2*x**6) - 15*d/x**8 - 21*d/x**10 - 21*d/x**12 - 15*d/x**14 - 15
*d/(2*x**16) - 5*d/(2*x**18) - d/(2*x**20) - d/(22*x**22) + e*log(x) - 5*e/x**2 - 45*e/(4*x**4) - 20*e/x**6 -
105*e/(4*x**8) - 126*e/(5*x**10) - 35*e/(2*x**12) - 60*e/(7*x**14) - 45*e/(16*x**16) - 5*e/(9*x**18) - e/(20*x
**20))/f**23, Eq(m, -23)), ((d*log(x) - 5*d/x**2 - 45*d/(4*x**4) - 20*d/x**6 - 105*d/(4*x**8) - 126*d/(5*x**10
) - 35*d/(2*x**12) - 60*d/(7*x**14) - 45*d/(16*x**16) - 5*d/(9*x**18) - d/(20*x**20) + e*x**2/2 + 10*e*log(x)
- 45*e/(2*x**2) - 30*e/x**4 - 35*e/x**6 - 63*e/(2*x**8) - 21*e/x**10 - 10*e/x**12 - 45*e/(14*x**14) - 5*e/(8*x
**16) - e/(18*x**18))/f**21, Eq(m, -21)), ((d*x**2/2 + 10*d*log(x) - 45*d/(2*x**2) - 30*d/x**4 - 35*d/x**6 - 6
3*d/(2*x**8) - 21*d/x**10 - 10*d/x**12 - 45*d/(14*x**14) - 5*d/(8*x**16) - d/(18*x**18) + e*x**4/4 + 5*e*x**2
+ 45*e*log(x) - 60*e/x**2 - 105*e/(2*x**4) - 42*e/x**6 - 105*e/(4*x**8) - 12*e/x**10 - 15*e/(4*x**12) - 5*e/(7
*x**14) - e/(16*x**16))/f**19, Eq(m, -19)), ((d*x**4/4 + 5*d*x**2 + 45*d*log(x) - 60*d/x**2 - 105*d/(2*x**4) -
42*d/x**6 - 105*d/(4*x**8) - 12*d/x**10 - 15*d/(4*x**12) - 5*d/(7*x**14) - d/(16*x**16) + e*x**6/6 + 5*e*x**4
/2 + 45*e*x**2/2 + 120*e*log(x) - 105*e/x**2 - 63*e/x**4 - 35*e/x**6 - 15*e/x**8 - 9*e/(2*x**10) - 5*e/(6*x**1
2) - e/(14*x**14))/f**17, Eq(m, -17)), ((d*x**6/6 + 5*d*x**4/2 + 45*d*x**2/2 + 120*d*log(x) - 105*d/x**2 - 63*
d/x**4 - 35*d/x**6 - 15*d/x**8 - 9*d/(2*x**10) - 5*d/(6*x**12) - d/(14*x**14) + e*x**8/8 + 5*e*x**6/3 + 45*e*x
**4/4 + 60*e*x**2 + 210*e*log(x) - 126*e/x**2 - 105*e/(2*x**4) - 20*e/x**6 - 45*e/(8*x**8) - e/x**10 - e/(12*x
**12))/f**15, Eq(m, -15)), ((d*x**8/8 + 5*d*x**6/3 + 45*d*x**4/4 + 60*d*x**2 + 210*d*log(x) - 126*d/x**2 - 105
*d/(2*x**4) - 20*d/x**6 - 45*d/(8*x**8) - d/x**10 - d/(12*x**12) + e*x**10/10 + 5*e*x**8/4 + 15*e*x**6/2 + 30*
e*x**4 + 105*e*x**2 + 252*e*log(x) - 105*e/x**2 - 30*e/x**4 - 15*e/(2*x**6) - 5*e/(4*x**8) - e/(10*x**10))/f**
13, Eq(m, -13)), ((d*x**10/10 + 5*d*x**8/4 + 15*d*x**6/2 + 30*d*x**4 + 105*d*x**2 + 252*d*log(x) - 105*d/x**2
- 30*d/x**4 - 15*d/(2*x**6) - 5*d/(4*x**8) - d/(10*x**10) + e*x**12/12 + e*x**10 + 45*e*x**8/8 + 20*e*x**6 + 1
05*e*x**4/2 + 126*e*x**2 + 210*e*log(x) - 60*e/x**2 - 45*e/(4*x**4) - 5*e/(3*x**6) - e/(8*x**8))/f**11, Eq(m,
-11)), ((d*x**12/12 + d*x**10 + 45*d*x**8/8 + 20*d*x**6 + 105*d*x**4/2 + 126*d*x**2 + 210*d*log(x) - 60*d/x**2
- 45*d/(4*x**4) - 5*d/(3*x**6) - d/(8*x**8) + e*x**14/14 + 5*e*x**12/6 + 9*e*x**10/2 + 15*e*x**8 + 35*e*x**6
+ 63*e*x**4 + 105*e*x**2 + 120*e*log(x) - 45*e/(2*x**2) - 5*e/(2*x**4) - e/(6*x**6))/f**9, Eq(m, -9)), ((d*x**
14/14 + 5*d*x**12/6 + 9*d*x**10/2 + 15*d*x**8 + 35*d*x**6 + 63*d*x**4 + 105*d*x**2 + 120*d*log(x) - 45*d/(2*x*
*2) - 5*d/(2*x**4) - d/(6*x**6) + e*x**16/16 + 5*e*x**14/7 + 15*e*x**12/4 + 12*e*x**10 + 105*e*x**8/4 + 42*e*x
**6 + 105*e*x**4/2 + 60*e*x**2 + 45*e*log(x) - 5*e/x**2 - e/(4*x**4))/f**7, Eq(m, -7)), ((d*x**16/16 + 5*d*x**
14/7 + 15*d*x**12/4 + 12*d*x**10 + 105*d*x**8/4 + 42*d*x**6 + 105*d*x**4/2 + 60*d*x**2 + 45*d*log(x) - 5*d/x**
2 - d/(4*x**4) + e*x**18/18 + 5*e*x**16/8 + 45*e*x**14/14 + 10*e*x**12 + 21*e*x**10 + 63*e*x**8/2 + 35*e*x**6
+ 30*e*x**4 + 45*e*x**2/2 + 10*e*log(x) - e/(2*x**2))/f**5, Eq(m, -5)), ((d*x**18/18 + 5*d*x**16/8 + 45*d*x**1
4/14 + 10*d*x**12 + 21*d*x**10 + 63*d*x**8/2 + 35*d*x**6 + 30*d*x**4 + 45*d*x**2/2 + 10*d*log(x) - d/(2*x**2)
+ e*x**20/20 + 5*e*x**18/9 + 45*e*x**16/16 + 60*e*x**14/7 + 35*e*x**12/2 + 126*e*x**10/5 + 105*e*x**8/4 + 20*e
*x**6 + 45*e*x**4/4 + 5*e*x**2 + e*log(x))/f**3, Eq(m, -3)), ((d*x**20/20 + 5*d*x**18/9 + 45*d*x**16/16 + 60*d
*x**14/7 + 35*d*x**12/2 + 126*d*x**10/5 + 105*d*x**8/4 + 20*d*x**6 + 45*d*x**4/4 + 5*d*x**2 + d*log(x) + e*x**
22/22 + e*x**20/2 + 5*e*x**18/2 + 15*e*x**16/2 + 15*e*x**14 + 21*e*x**12 + 21*e*x**10 + 15*e*x**8 + 15*e*x**6/
2 + 5*e*x**4/2 + e*x**2/2)/f, Eq(m, -1)), (d*m**11*x**21*(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 + 264300628944*m**3
+ 590546123298*m**2 + 703416314160*m + 316234143225) + 10*d*m**11*x**19*(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 + 264
300628944*m**3 + 590546123298*m**2 + 703416314160*m + 316234143225) + 45*d*m**11*x**17*(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 + 7257825
9391*m**4 + 264300628944*m**3 + 590546123298*m**2 + 703416314160*m + 316234143225) + 120*d*m**11*x**15*(f*x)**
m/(m**12 + 144*m**11 + 9218*m**10 + 345840*m**9 + 8439783*m**8 + 140529312*m**7 + 1628301884*m**6 + 1313745840
0*m**5 + 72578259391*m**4 + 264300628944*m**3 + 590546123298*m**2 + 703416314160*m + 316234143225) + 210*d*m**
11*x**13*(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 + 264...
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3752 vs.
\(2 (280) = 560\).
time = 3.97, size = 3752, normalized size = 13.95 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x)^m*(e*x^2+d)*(x^4+2*x^2+1)^5,x, algorithm="giac")
[Out]
((f*x)^m*m^11*x^23*e + 121*(f*x)^m*m^10*x^23*e + (f*x)^m*d*m^11*x^21 + 10*(f*x)^m*m^11*x^21*e + 6435*(f*x)^m*m
^9*x^23*e + 123*(f*x)^m*d*m^10*x^21 + 1230*(f*x)^m*m^10*x^21*e + 197835*(f*x)^m*m^8*x^23*e + 10*(f*x)^m*d*m^11
*x^19 + 6635*(f*x)^m*d*m^9*x^21 + 45*(f*x)^m*m^11*x^19*e + 66350*(f*x)^m*m^9*x^21*e + 3889578*(f*x)^m*m^7*x^23
*e + 1250*(f*x)^m*d*m^10*x^19 + 206505*(f*x)^m*d*m^8*x^21 + 5625*(f*x)^m*m^10*x^19*e + 2065050*(f*x)^m*m^8*x^2
1*e + 51069018*(f*x)^m*m^6*x^23*e + 45*(f*x)^m*d*m^11*x^17 + 68430*(f*x)^m*d*m^9*x^19 + 4103178*(f*x)^m*d*m^7*
x^21 + 120*(f*x)^m*m^11*x^17*e + 307935*(f*x)^m*m^9*x^19*e + 41031780*(f*x)^m*m^7*x^21*e + 453714470*(f*x)^m*m
^5*x^23*e + 5715*(f*x)^m*d*m^10*x^17 + 2158230*(f*x)^m*d*m^8*x^19 + 54362574*(f*x)^m*d*m^6*x^21 + 15240*(f*x)^
m*m^10*x^17*e + 9712035*(f*x)^m*m^8*x^19*e + 543625740*(f*x)^m*m^6*x^21*e + 2702025590*(f*x)^m*m^4*x^23*e + 12
0*(f*x)^m*d*m^11*x^15 + 317655*(f*x)^m*d*m^9*x^17 + 43391460*(f*x)^m*d*m^7*x^19 + 486687830*(f*x)^m*d*m^5*x^21
+ 210*(f*x)^m*m^11*x^15*e + 847080*(f*x)^m*m^9*x^17*e + 195261570*(f*x)^m*m^7*x^19*e + 4866878300*(f*x)^m*m^5
*x^21*e + 10431670821*(f*x)^m*m^3*x^23*e + 15480*(f*x)^m*d*m^10*x^15 + 10162665*(f*x)^m*d*m^8*x^17 + 580855380
*(f*x)^m*d*m^6*x^19 + 2917013970*(f*x)^m*d*m^4*x^21 + 27090*(f*x)^m*m^10*x^15*e + 27100440*(f*x)^m*m^8*x^17*e
+ 2613849210*(f*x)^m*m^6*x^19*e + 29170139700*(f*x)^m*m^4*x^21*e + 24372200061*(f*x)^m*m^2*x^23*e + 210*(f*x)^
m*d*m^11*x^13 + 873960*(f*x)^m*d*m^9*x^15 + 207024930*(f*x)^m*d*m^7*x^17 + 5246766620*(f*x)^m*d*m^5*x^19 + 113
20966021*(f*x)^m*d*m^3*x^21 + 252*(f*x)^m*m^11*x^13*e + 1529430*(f*x)^m*m^9*x^15*e + 552066480*(f*x)^m*m^7*x^1
7*e + 23610449790*(f*x)^m*m^5*x^19*e + 113209660210*(f*x)^m*m^3*x^21*e + 29985521895*(f*x)^m*m*x^23*e + 27510*
(f*x)^m*d*m^10*x^13 + 28391400*(f*x)^m*d*m^8*x^15 + 2804395230*(f*x)^m*d*m^6*x^17 + 31686018220*(f*x)^m*d*m^4*
x^19 + 26560342503*(f*x)^m*d*m^2*x^21 + 33012*(f*x)^m*m^10*x^13*e + 49684950*(f*x)^m*m^8*x^15*e + 7478387280*(
f*x)^m*m^6*x^17*e + 142587081990*(f*x)^m*m^4*x^19*e + 265603425030*(f*x)^m*m^2*x^21*e + 13749310575*(f*x)^m*x^
23*e + 252*(f*x)^m*d*m^11*x^11 + 1578150*(f*x)^m*d*m^9*x^13 + 586902960*(f*x)^m*d*m^7*x^15 + 25598865870*(f*x)
^m*d*m^5*x^17 + 123748247730*(f*x)^m*d*m^3*x^19 + 32778930735*(f*x)^m*d*m*x^21 + 210*(f*x)^m*m^11*x^11*e + 189
3780*(f*x)^m*m^9*x^13*e + 1027080180*(f*x)^m*m^7*x^15*e + 68263642320*(f*x)^m*m^5*x^17*e + 556867114785*(f*x)^
m*m^3*x^19*e + 327789307350*(f*x)^m*m*x^21*e + 33516*(f*x)^m*d*m^10*x^11 + 52110450*(f*x)^m*d*m^8*x^13 + 80599
73040*(f*x)^m*d*m^6*x^15 + 156004908210*(f*x)^m*d*m^4*x^17 + 291789582570*(f*x)^m*d*m^2*x^19 + 15058768725*(f*
x)^m*d*x^21 + 27930*(f*x)^m*m^10*x^11*e + 62532540*(f*x)^m*m^8*x^13*e + 14104952820*(f*x)^m*m^6*x^15*e + 41601
3088560*(f*x)^m*m^4*x^17*e + 1313053121565*(f*x)^m*m^2*x^19*e + 150587687250*(f*x)^m*x^21*e + 210*(f*x)^m*d*m^
11*x^9 + 1954260*(f*x)^m*d*m^9*x^11 + 1094918580*(f*x)^m*d*m^7*x^13 + 74496630480*(f*x)^m*d*m^5*x^15 + 6139382
33025*(f*x)^m*d*m^3*x^17 + 361459164150*(f*x)^m*d*m*x^19 + 120*(f*x)^m*m^11*x^9*e + 1628550*(f*x)^m*m^9*x^11*e
+ 1313902296*(f*x)^m*m^7*x^13*e + 130369103340*(f*x)^m*m^5*x^15*e + 1637168621400*(f*x)^m*m^3*x^17*e + 162656
6238675*(f*x)^m*m*x^19*e + 28350*(f*x)^m*d*m^10*x^9 + 65654820*(f*x)^m*d*m^8*x^11 + 15277213980*(f*x)^m*d*m^6*
x^13 + 459045550800*(f*x)^m*d*m^4*x^15 + 1456578341055*(f*x)^m*d*m^2*x^17 + 166439022750*(f*x)^m*d*x^19 + 1620
0*(f*x)^m*m^10*x^9*e + 54712350*(f*x)^m*m^8*x^11*e + 18332656776*(f*x)^m*m^6*x^13*e + 803329713900*(f*x)^m*m^4
*x^15*e + 3884208909480*(f*x)^m*m^2*x^17*e + 748975602375*(f*x)^m*x^19*e + 120*(f*x)^m*d*m^11*x^7 + 1680630*(f
*x)^m*d*m^9*x^9 + 1404622296*(f*x)^m*d*m^7*x^11 + 143339613900*(f*x)^m*d*m^5*x^13 + 1823707864920*(f*x)^m*d*m^
3*x^15 + 1812743750475*(f*x)^m*d*m*x^17 + 45*(f*x)^m*m^11*x^7*e + 960360*(f*x)^m*m^9*x^9*e + 1170518580*(f*x)^
m*m^7*x^11*e + 172007536680*(f*x)^m*m^5*x^13*e + 3191488763610*(f*x)^m*m^3*x^15*e + 4833983334600*(f*x)^m*m*x^
17*e + 16440*(f*x)^m*d*m^10*x^7 + 57500730*(f*x)^m*d*m^8*x^9 + 19962541368*(f*x)^m*d*m^6*x^11 + 895451283300*(
f*x)^m*d*m^4*x^13 + 4360457499480*(f*x)^m*d*m^2*x^15 + 837090379125*(f*x)^m*d*x^17 + 6165*(f*x)^m*m^10*x^7*e +
32857560*(f*x)^m*m^8*x^9*e + 16635451140*(f*x)^m*m^6*x^11*e + 1074541539960*(f*x)^m*m^4*x^13*e + 763080062409
0*(f*x)^m*m^2*x^15*e + 2232241011000*(f*x)^m*x^17*e + 45*(f*x)^m*d*m^11*x^5 + 991080*(f*x)^m*d*m^9*x^7 + 12548
47860*(f*x)^m*d*m^7*x^9 + 190744119720*(f*x)^m*d*m^5*x^11 + 3600567789210*(f*x)^m*d*m^3*x^13 + 5458672303560*(
f*x)^m*d*m*x^15 + 10*(f*x)^m*m^11*x^5*e + 371655*(f*x)^m*m^9*x^7*e + 717055920*(f*x)^m*m^7*x^9*e + 15895343310
0*(f*x)^m*m^5*x^11*e + 4320681347052*(f*x)^m*m^3*x^13*e + 9552676531230*(f*x)^m*m*x^15*e + 6255*(f*x)^m*d*m^10
*x^5 + 34563240*(f*x)^m*d*m^8*x^7 + 18217524780*(f*x)^m*d*m^6*x^9 + 1212454199880*(f*x)^m*d*m^4*x^11 + 8695750
818510*(f*x)^m*d*m^2*x^13 + 2529873145800*(f*x)^m*d*x^15 + 1390*(f*x)^m*m^10*x^5*e + 12961215*(f*x)^m*m^8*x^7*
e + 10410014160*(f*x)^m*m^6*x^9*e + 10103784999...
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Mupad [B]
time = 1.78, size = 1539, normalized size = 5.72 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x)^m*(d + e*x^2)*(2*x^2 + x^4 + 1)^5,x)
[Out]
(d*x*(f*x)^m*(387182170935*m + 203363952363*m^2 + 60936676581*m^3 + 11641582810*m^4 + 1495875590*m^5 + 1324262
94*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 + 3
45840*m^9 + 9218*m^10 + 144*m^11 + m^12 + 316234143225) + (e*x^23*(f*x)^m*(29985521895*m + 24372200061*m^2 + 1
0431670821*m^3 + 2702025590*m^4 + 453714470*m^5 + 51069018*m^6 + 3889578*m^7 + 197835*m^8 + 6435*m^9 + 121*m^1
0 + 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)
+ (30*x^15*(f*x)^m*(4*d + 7*e)*(45488935863*m + 36337145829*m^2 + 15197565541*m^3 + 3825379590*m^4 + 62080525
4*m^5 + 67166442*m^6 + 4890858*m^7 + 236595*m^8 + 7283*m^9 + 129*m^10 + m^11 + 21082276215))/(703416314160*m +
590546123298*m^2 + 264300628944*m^3 + 72578259391*m^4 + 13137458400*m^5 + 1628301884*m^6 + 140529312*m^7 + 84
39783*m^8 + 345840*m^9 + 9218*m^10 + 144*m^11 + m^12 + 316234143225) + (42*x^13*(f*x)^m*(5*d + 6*e)*(522377392
95*m + 41408337231*m^2 + 17145560901*m^3 + 4264053730*m^4 + 682569590*m^5 + 72748638*m^6 + 5213898*m^7 + 24814
5*m^8 + 7515*m^9 + 131*m^10 + m^11 + 24325703325))/(703416314160*m + 590546123298*m^2 + 264300628944*m^3 + 725
78259391*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) + (30*x^9*(f*x)^m*(7*d + 4*e)*(74253243015*m + 57365875587*m^2 + 22992750373*m^3 +
5509501002*m^4 + 847550822*m^5 + 86750118*m^6 + 5975466*m^7 + 273813*m^8 + 8003*m^9 + 135*m^10 + m^11 + 351371
27025))/(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^3*(f*x)^m*(
10*d + e)*(199334977695*m + 130403715201*m^2 + 44632304581*m^3 + 9315318270*m^4 + 1274046710*m^5 + 118085058*m
^6 + 7481418*m^7 + 319455*m^8 + 8795*m^9 + 141*m^10 + m^11 + 105411381075))/(703416314160*m + 590546123298*m^2
+ 264300628944*m^3 + 72578259391*m^4 + 13137458400*m^5 + 1628301884*m^6 + 140529312*m^7 + 8439783*m^8 + 34584
0*m^9 + 9218*m^10 + 144*m^11 + m^12 + 316234143225) + (5*x^19*(f*x)^m*(2*d + 9*e)*(36145916415*m + 29178958257
*m^2 + 12374824773*m^3 + 3168601822*m^4 + 524676662*m^5 + 58085538*m^6 + 4339146*m^7 + 215823*m^8 + 6843*m^9 +
125*m^10 + m^11 + 16643902275))/(703416314160*m + 590546123298*m^2 + 264300628944*m^3 + 72578259391*m^4 + 131
37458400*m^5 + 1628301884*m^6 + 140529312*m^7 + 8439783*m^8 + 345840*m^9 + 9218*m^10 + 144*m^11 + m^12 + 31623
4143225) + (42*x^11*(f*x)^m*(6*d + 5*e)*(61333432335*m + 48110244633*m^2 + 19653671301*m^3 + 4811326190*m^4 +
756921110*m^5 + 79216434*m^6 + 5573898*m^7 + 260535*m^8 + 7755*m^9 + 133*m^10 + m^11 + 28748558475))/(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) + (15*x^7*(f*x)^m*(8*d + 3*e)*(94
034286855*m + 70930262349*m^2 + 27624338085*m^3 + 6421988758*m^4 + 959352806*m^5 + 95564154*m^6 + 6423594*m^7
+ 288027*m^8 + 8259*m^9 + 137*m^10 + m^11 + 45176306175))/(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) + (5*x^5*(f*x)^m*(9*d + 2*e)*(128033897103*m + 92502445239*m^2 + 34359636741
*m^3 + 7643724530*m^4 + 1098746774*m^5 + 105911022*m^6 + 6923658*m^7 + 303225*m^8 + 8523*m^9 + 139*m^10 + m^11
+ 63246828645))/(703416314160*m + 590546123298*m^2 + 264300628944*m^3 + 72578259391*m^4 + 13137458400*m^5 + 1
628301884*m^6 + 140529312*m^7 + 8439783*m^8 + 345840*m^9 + 9218*m^10 + 144*m^11 + m^12 + 316234143225) + (15*x
^17*(f*x)^m*(3*d + 8*e)*(40283194455*m + 32368407579*m^2 + 13643071845*m^3 + 3466775738*m^4 + 568863686*m^5 +
62319894*m^6 + 4600554*m^7 + 225837*m^8 + 7059*m^9 + 127*m^10 + m^11 + 18602008425))/(703416314160*m + 5905461
23298*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^21*(f*x)^m*(d + 10*e)*(32778930735*m + 26560
342503*m^2 + 11320966021*m^3 + 2917013970*m^4 + 486687830*m^5 + 54362574*m^6 + 4103178*m^7 + 206505*m^8 + 6635
*m^9 + 123*m^10 + m^11 + 15058768725))/(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)
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