3.55 \(\int (f x)^m (d+e x^2) (1+2 x^2+x^4)^5 \, dx\)
Optimal. Leaf size=269 \[ \frac{(10 d+e) (f x)^{m+3}}{f^3 (m+3)}+\frac{5 (9 d+2 e) (f x)^{m+5}}{f^5 (m+5)}+\frac{15 (8 d+3 e) (f x)^{m+7}}{f^7 (m+7)}+\frac{30 (7 d+4 e) (f x)^{m+9}}{f^9 (m+9)}+\frac{42 (6 d+5 e) (f x)^{m+11}}{f^{11} (m+11)}+\frac{42 (5 d+6 e) (f x)^{m+13}}{f^{13} (m+13)}+\frac{30 (4 d+7 e) (f x)^{m+15}}{f^{15} (m+15)}+\frac{15 (3 d+8 e) (f x)^{m+17}}{f^{17} (m+17)}+\frac{5 (2 d+9 e) (f x)^{m+19}}{f^{19} (m+19)}+\frac{(d+10 e) (f x)^{m+21}}{f^{21} (m+21)}+\frac{d (f x)^{m+1}}{f (m+1)}+\frac{e (f x)^{m+23}}{f^{23} (m+23)} \]
[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 = 0.160783, antiderivative size = 269, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used =
{28, 448} \[ \frac{(10 d+e) (f x)^{m+3}}{f^3 (m+3)}+\frac{5 (9 d+2 e) (f x)^{m+5}}{f^5 (m+5)}+\frac{15 (8 d+3 e) (f x)^{m+7}}{f^7 (m+7)}+\frac{30 (7 d+4 e) (f x)^{m+9}}{f^9 (m+9)}+\frac{42 (6 d+5 e) (f x)^{m+11}}{f^{11} (m+11)}+\frac{42 (5 d+6 e) (f x)^{m+13}}{f^{13} (m+13)}+\frac{30 (4 d+7 e) (f x)^{m+15}}{f^{15} (m+15)}+\frac{15 (3 d+8 e) (f x)^{m+17}}{f^{17} (m+17)}+\frac{5 (2 d+9 e) (f x)^{m+19}}{f^{19} (m+19)}+\frac{(d+10 e) (f x)^{m+21}}{f^{21} (m+21)}+\frac{d (f x)^{m+1}}{f (m+1)}+\frac{e (f x)^{m+23}}{f^{23} (m+23)} \]
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 448
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*}
Mathematica [A] time = 0.170736, size = 189, normalized size = 0.7 \[ x (f x)^m \left (\frac{x^{20} (d+10 e)}{m+21}+\frac{5 x^{18} (2 d+9 e)}{m+19}+\frac{15 x^{16} (3 d+8 e)}{m+17}+\frac{30 x^{14} (4 d+7 e)}{m+15}+\frac{42 x^{12} (5 d+6 e)}{m+13}+\frac{42 x^{10} (6 d+5 e)}{m+11}+\frac{30 x^8 (7 d+4 e)}{m+9}+\frac{15 x^6 (8 d+3 e)}{m+7}+\frac{5 x^4 (9 d+2 e)}{m+5}+\frac{x^2 (10 d+e)}{m+3}+\frac{d}{m+1}+\frac{e x^{22}}{m+23}\right ) \]
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] time = 0.019, size = 2295, normalized size = 8.5 \begin{align*} \text{result too large to display} \end{align*}
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)
[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^4+2032933777875*e*x^6+1495875590*d*m^5+446323045810*d*m^3*x^2+
5761525369635*d*m*x^4+44632304581*e*m^3*x^2+1280338971030*e*m*x^4+11641582810*d*m^4+1304037152010*d*m^2*x^2+28
46107289025*d*x^4+130403715201*e*m^2*x^2+632468286450*e*x^4+60936676581*d*m^3+1993349776950*d*m*x^2+1993349776
95*e*m*x^2+203363952363*d*m^2+1054113810750*d*x^2+105411381075*e*x^2+387182170935*d*m+316234143225*d)*x/(1+m)/
(3+m)/(5+m)/(7+m)/(9+m)/(11+m)/(13+m)/(15+m)/(17+m)/(19+m)/(21+m)/(23+m)
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
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]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 1.77115, size = 5265, normalized size = 19.57 \begin{align*} \text{result too large to display} \end{align*}
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]
((e*m^11 + 121*e*m^10 + 6435*e*m^9 + 197835*e*m^8 + 3889578*e*m^7 + 51069018*e*m^6 + 453714470*e*m^5 + 2702025
590*e*m^4 + 10431670821*e*m^3 + 24372200061*e*m^2 + 29985521895*e*m + 13749310575*e)*x^23 + ((d + 10*e)*m^11 +
123*(d + 10*e)*m^10 + 6635*(d + 10*e)*m^9 + 206505*(d + 10*e)*m^8 + 4103178*(d + 10*e)*m^7 + 54362574*(d + 10
*e)*m^6 + 486687830*(d + 10*e)*m^5 + 2917013970*(d + 10*e)*m^4 + 11320966021*(d + 10*e)*m^3 + 26560342503*(d +
10*e)*m^2 + 32778930735*(d + 10*e)*m + 15058768725*d + 150587687250*e)*x^21 + 5*((2*d + 9*e)*m^11 + 125*(2*d
+ 9*e)*m^10 + 6843*(2*d + 9*e)*m^9 + 215823*(2*d + 9*e)*m^8 + 4339146*(2*d + 9*e)*m^7 + 58085538*(2*d + 9*e)*m
^6 + 524676662*(2*d + 9*e)*m^5 + 3168601822*(2*d + 9*e)*m^4 + 12374824773*(2*d + 9*e)*m^3 + 29178958257*(2*d +
9*e)*m^2 + 36145916415*(2*d + 9*e)*m + 33287804550*d + 149795120475*e)*x^19 + 15*((3*d + 8*e)*m^11 + 127*(3*d
+ 8*e)*m^10 + 7059*(3*d + 8*e)*m^9 + 225837*(3*d + 8*e)*m^8 + 4600554*(3*d + 8*e)*m^7 + 62319894*(3*d + 8*e)*
m^6 + 568863686*(3*d + 8*e)*m^5 + 3466775738*(3*d + 8*e)*m^4 + 13643071845*(3*d + 8*e)*m^3 + 32368407579*(3*d
+ 8*e)*m^2 + 40283194455*(3*d + 8*e)*m + 55806025275*d + 148816067400*e)*x^17 + 30*((4*d + 7*e)*m^11 + 129*(4*
d + 7*e)*m^10 + 7283*(4*d + 7*e)*m^9 + 236595*(4*d + 7*e)*m^8 + 4890858*(4*d + 7*e)*m^7 + 67166442*(4*d + 7*e)
*m^6 + 620805254*(4*d + 7*e)*m^5 + 3825379590*(4*d + 7*e)*m^4 + 15197565541*(4*d + 7*e)*m^3 + 36337145829*(4*d
+ 7*e)*m^2 + 45488935863*(4*d + 7*e)*m + 84329104860*d + 147575933505*e)*x^15 + 42*((5*d + 6*e)*m^11 + 131*(5
*d + 6*e)*m^10 + 7515*(5*d + 6*e)*m^9 + 248145*(5*d + 6*e)*m^8 + 5213898*(5*d + 6*e)*m^7 + 72748638*(5*d + 6*e
)*m^6 + 682569590*(5*d + 6*e)*m^5 + 4264053730*(5*d + 6*e)*m^4 + 17145560901*(5*d + 6*e)*m^3 + 41408337231*(5*
d + 6*e)*m^2 + 52237739295*(5*d + 6*e)*m + 121628516625*d + 145954219950*e)*x^13 + 42*((6*d + 5*e)*m^11 + 133*
(6*d + 5*e)*m^10 + 7755*(6*d + 5*e)*m^9 + 260535*(6*d + 5*e)*m^8 + 5573898*(6*d + 5*e)*m^7 + 79216434*(6*d + 5
*e)*m^6 + 756921110*(6*d + 5*e)*m^5 + 4811326190*(6*d + 5*e)*m^4 + 19653671301*(6*d + 5*e)*m^3 + 48110244633*(
6*d + 5*e)*m^2 + 61333432335*(6*d + 5*e)*m + 172491350850*d + 143742792375*e)*x^11 + 30*((7*d + 4*e)*m^11 + 13
5*(7*d + 4*e)*m^10 + 8003*(7*d + 4*e)*m^9 + 273813*(7*d + 4*e)*m^8 + 5975466*(7*d + 4*e)*m^7 + 86750118*(7*d +
4*e)*m^6 + 847550822*(7*d + 4*e)*m^5 + 5509501002*(7*d + 4*e)*m^4 + 22992750373*(7*d + 4*e)*m^3 + 57365875587
*(7*d + 4*e)*m^2 + 74253243015*(7*d + 4*e)*m + 245959889175*d + 140548508100*e)*x^9 + 15*((8*d + 3*e)*m^11 + 1
37*(8*d + 3*e)*m^10 + 8259*(8*d + 3*e)*m^9 + 288027*(8*d + 3*e)*m^8 + 6423594*(8*d + 3*e)*m^7 + 95564154*(8*d
+ 3*e)*m^6 + 959352806*(8*d + 3*e)*m^5 + 6421988758*(8*d + 3*e)*m^4 + 27624338085*(8*d + 3*e)*m^3 + 7093026234
9*(8*d + 3*e)*m^2 + 94034286855*(8*d + 3*e)*m + 361410449400*d + 135528918525*e)*x^7 + 5*((9*d + 2*e)*m^11 + 1
39*(9*d + 2*e)*m^10 + 8523*(9*d + 2*e)*m^9 + 303225*(9*d + 2*e)*m^8 + 6923658*(9*d + 2*e)*m^7 + 105911022*(9*d
+ 2*e)*m^6 + 1098746774*(9*d + 2*e)*m^5 + 7643724530*(9*d + 2*e)*m^4 + 34359636741*(9*d + 2*e)*m^3 + 92502445
239*(9*d + 2*e)*m^2 + 128033897103*(9*d + 2*e)*m + 569221457805*d + 126493657290*e)*x^5 + ((10*d + e)*m^11 + 1
41*(10*d + e)*m^10 + 8795*(10*d + e)*m^9 + 319455*(10*d + e)*m^8 + 7481418*(10*d + e)*m^7 + 118085058*(10*d +
e)*m^6 + 1274046710*(10*d + e)*m^5 + 9315318270*(10*d + e)*m^4 + 44632304581*(10*d + e)*m^3 + 130403715201*(10
*d + e)*m^2 + 199334977695*(10*d + e)*m + 1054113810750*d + 105411381075*e)*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 + 203363952363*d*m^2 + 387182170935*d*m + 316234143225*d)*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 + 264300628944*m^3 + 5
90546123298*m^2 + 703416314160*m + 316234143225)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.30771, size = 5065, normalized size = 18.83 \begin{align*} \text{result too large to display} \end{align*}
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 + 1010378499900*(f*x)^m*m^4*x^11*e + 10434900982212*(f*x)^m*m^2*x^13*e + 442
7278005150*(f*x)^m*x^15*e + 10*(f*x)^m*d*m^11*x^3 + 383535*(f*x)^m*d*m^9*x^5 + 770831280*(f*x)^m*d*m^7*x^7 + 1
77985672620*(f*x)^m*d*m^5*x^9 + 4952725167852*(f*x)^m*d*m^3*x^11 + 10969925251950*(f*x)^m*d*m*x^13 + (f*x)^m*m
^11*x^3*e + 85230*(f*x)^m*m^9*x^5*e + 289061730*(f*x)^m*m^7*x^7*e + 101706098640*(f*x)^m*m^5*x^9*e + 412727097
3210*(f*x)^m*m^3*x^11*e + 13163910302340*(f*x)^m*m*x^13*e + 1410*(f*x)^m*d*m^10*x^3 + 13645125*(f*x)^m*d*m^8*x
^5 + 11467698480*(f*x)^m*d*m^6*x^7 + 1156995210420*(f*x)^m*d*m^4*x^9 + 12123781647516*(f*x)^m*d*m^2*x^11 + 510
8397698250*(f*x)^m*d*x^13 + 141*(f*x)^m*m^10*x^3*e + 3032250*(f*x)^m*m^8*x^5*e + 4300386930*(f*x)^m*m^6*x^7*e
+ 661140120240*(f*x)^m*m^4*x^9*e + 10103151372930*(f*x)^m*m^2*x^11*e + 6130077237900*(f*x)^m*x^13*e + (f*x)^m*
d*m^11*x + 87950*(f*x)^m*d*m^9*x^3 + 311564610*(f*x)^m*d*m^7*x^5 + 115122336720*(f*x)^m*d*m^5*x^7 + 4828477578
330*(f*x)^m*d*m^3*x^9 + 15456024948420*(f*x)^m*d*m*x^11 + 8795*(f*x)^m*m^9*x^3*e + 69236580*(f*x)^m*m^7*x^5*e
+ 43170876270*(f*x)^m*m^5*x^7*e + 2759130044760*(f*x)^m*m^3*x^9*e + 12880020790350*(f*x)^m*m*x^11*e + 143*(f*x
)^m*d*m^10*x + 3194550*(f*x)^m*d*m^8*x^3 + 4765995990*(f*x)^m*d*m^6*x^5 + 770638650960*(f*x)^m*d*m^4*x^7 + 120
46833873270*(f*x)^m*d*m^2*x^9 + 7244636735700*(f*x)^m*d*x^11 + 319455*(f*x)^m*m^8*x^3*e + 1059110220*(f*x)^m*m
^6*x^5*e + 288989494110*(f*x)^m*m^4*x^7*e + 6883905070440*(f*x)^m*m^2*x^9*e + 6037197279750*(f*x)^m*x^11*e + 9
075*(f*x)^m*d*m^9*x + 74814180*(f*x)^m*d*m^7*x^3 + 49443604830*(f*x)^m*d*m^5*x^5 + 3314920570200*(f*x)^m*d*m^3
*x^7 + 15593181033150*(f*x)^m*d*m*x^9 + 7481418*(f*x)^m*m^7*x^3*e + 10987467740*(f*x)^m*m^5*x^5*e + 1243095213
825*(f*x)^m*m^3*x^7*e + 8910389161800*(f*x)^m*m*x^9*e + 336765*(f*x)^m*d*m^8*x + 1180850580*(f*x)^m*d*m^6*x^3
+ 343967603850*(f*x)^m*d*m^4*x^5 + 8511631481880*(f*x)^m*d*m^2*x^7 + 7378796675250*(f*x)^m*d*x^9 + 118085058*(
f*x)^m*m^6*x^3*e + 76437245300*(f*x)^m*m^4*x^5*e + 3191861805705*(f*x)^m*m^2*x^7*e + 4216455243000*(f*x)^m*x^9
*e + 8103018*(f*x)^m*d*m^7*x + 12740467100*(f*x)^m*d*m^5*x^3 + 1546183653345*(f*x)^m*d*m^3*x^5 + 1128411442260
0*(f*x)^m*d*m*x^7 + 1274046710*(f*x)^m*m^5*x^3*e + 343596367410*(f*x)^m*m^3*x^5*e + 4231542908475*(f*x)^m*m*x^
7*e + 132426294*(f*x)^m*d*m^6*x + 93153182700*(f*x)^m*d*m^4*x^3 + 4162610035755*(f*x)^m*d*m^2*x^5 + 5421156741
000*(f*x)^m*d*x^7 + 9315318270*(f*x)^m*m^4*x^3*e + 925024452390*(f*x)^m*m^2*x^5*e + 2032933777875*(f*x)^m*x^7*
e + 1495875590*(f*x)^m*d*m^5*x + 446323045810*(f*x)^m*d*m^3*x^3 + 5761525369635*(f*x)^m*d*m*x^5 + 44632304581*
(f*x)^m*m^3*x^3*e + 1280338971030*(f*x)^m*m*x^5*e + 11641582810*(f*x)^m*d*m^4*x + 1304037152010*(f*x)^m*d*m^2*
x^3 + 2846107289025*(f*x)^m*d*x^5 + 130403715201*(f*x)^m*m^2*x^3*e + 632468286450*(f*x)^m*x^5*e + 60936676581*
(f*x)^m*d*m^3*x + 1993349776950*(f*x)^m*d*m*x^3 + 199334977695*(f*x)^m*m*x^3*e + 203363952363*(f*x)^m*d*m^2*x
+ 1054113810750*(f*x)^m*d*x^3 + 105411381075*(f*x)^m*x^3*e + 387182170935*(f*x)^m*d*m*x + 316234143225*(f*x)^m
*d*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)