∫ −16588800x−30965760x3−24883200x5−10936320x7−2744320x9−368640x11−20480x13+e4(3110400x+10414080x3+12960000x5+8542720x7+3310080x9+760320x11+96000x13+5120x15)____________________________________________________________________________________________________________________________________ 19683+126846x2+331533x4+457552x6+369117x8+184710x10+58563x12+11532x14+1296x16+64x18 dx

3.21.58 \(\int \frac {-16588800 x-30965760 x^3-24883200 x^5-10936320 x^7-2744320 x^9-368640 x^{11}-20480 x^{13}+e^4 (3110400 x+10414080 x^3+12960000 x^5+8542720 x^7+3310080 x^9+760320 x^{11}+96000 x^{13}+5120 x^{15})}{19683+126846 x^2+331533 x^4+457552 x^6+369117 x^8+184710 x^{10}+58563 x^{12}+11532 x^{14}+1296 x^{16}+64 x^{18}} \, dx\)

Optimal. Leaf size=32 \[ 5 \left (-e^4+\frac {4}{\frac {3}{4}+x^2+\frac {x^2}{\left (3+x^2\right )^2}}\right )^2 \]

________________________________________________________________________________________

Rubi [C] time = 175.79, antiderivative size = 10556, normalized size of antiderivative = 329.88, number of steps used = 36, number of rules used = 12, integrand size = 127, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2073, 2101, 1593, 6715, 2081, 2079, 822, 800, 634, 618, 204, 628}

result too large to display

Warning: Unable to verify antiderivative.

[In]

Int[(-16588800*x - 30965760*x^3 - 24883200*x^5 - 10936320*x^7 - 2744320*x^9 - 368640*x^11 - 20480*x^13 + E^4*(

3110400*x + 10414080*x^3 + 12960000*x^5 + 8542720*x^7 + 3310080*x^9 + 760320*x^11 + 96000*x^13 + 5120*x^15))/(

19683 + 126846*x^2 + 331533*x^4 + 457552*x^6 + 369117*x^8 + 184710*x^10 + 58563*x^12 + 11532*x^14 + 1296*x^16

+ 64*x^18),x]

[Out]

44005/(3*(27 + 58*x^2 + 27*x^4 + 4*x^6)^2) + (10*(288 - 49*E^4))/(3*(27 + 58*x^2 + 27*x^4 + 4*x^6)) - (1865683

860480*(81 + 8*Sqrt[102])^(2/3)*(5*3^(1/6)*11^(2/3)*(152080197*Sqrt[3] + 45174544*Sqrt[34])*(81 - 8*Sqrt[102])

^(1/3) - 6*3^(1/6)*11^(2/3)*(2158158181*Sqrt[3] + 640900564*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 349030*(81 +

8*Sqrt[102])^(2/3)*(4363 + 432*Sqrt[102]) + 3840*33^(1/3)*(153347567 + 15183672*Sqrt[102]) - 3*(81 - 8*Sqrt[10

2])^(2/3)*(370863132822 + 36720944888*Sqrt[102] + 2742209545*(33*(81 + 8*Sqrt[102]))^(1/3) + 271519200*3^(5/6)

*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3))))/((33*33^(1/3) + 3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqr

t[102])^(1/3) + 33*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sqrt[102])^(2/3)*(8726 + 864*Sqrt[102]))*(3*(33*33^(1/3)

+ 4*3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) - 44*(81 + 8*Sqrt[102])^(2/3)) - (81

- 8*Sqrt[102])^(2/3)*(16*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3) + 3*(4363 + 432*Sqrt[102] + 54*(33*(81 +

8*Sqrt[102]))^(1/3))))^2*(99*33^(1/3) + 3*3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3)

- 33*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sqrt[102])^(2/3)*(32*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3) + 6*(

4363 + 432*Sqrt[102] + 54*(33*(81 + 8*Sqrt[102]))^(1/3))))^2*((81 - 8*Sqrt[102])^(1/3)*(33^(1/3) + (81 + 8*Sqr

t[102])^(2/3)) - 3*(9 + 4*x^2))^2) - (2035291484160*(81 + 8*Sqrt[102])^(1/3)*(501552*3^(1/3)*(47993 - 4752*Sqr

t[102])^(2/3)*(384502464 + 38071417*Sqrt[102]) - 11*(4*3^(1/6)*11^(2/3)*(9989518737005*Sqrt[3] + 2967329230258

*Sqrt[34])*(81 - 8*Sqrt[102])^(1/3) + 11034144*33^(1/3)*(44064 + 4363*Sqrt[102]) - (81 + 8*Sqrt[102])^(1/3)*(9

824407526579*33^(2/3) + 2918283938024*3^(1/6)*11^(2/3)*Sqrt[34] + 708106160160*3^(1/6)*Sqrt[34]*(4363 - 432*Sq

rt[102])^(2/3) + 2383840544256*(3*(4363 - 432*Sqrt[102]))^(2/3)) - (81 + 8*Sqrt[102])^(2/3)*(940336291345 + 93

107164128*Sqrt[102] - 956513005200*3^(5/6)*Sqrt[34]*(891 - 88*Sqrt[102])^(1/3) - 9660304030891*(33*(81 - 8*Sqr

t[102]))^(1/3)) + (81 - 8*Sqrt[102])^(2/3)*(26551002050056 + 2628940870952*Sqrt[102] + 1150139003149*(33*(81 +

8*Sqrt[102]))^(1/3) + 113880730752*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3)))))/((33*33^(1/3) + 3^(1/6)*11

^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 33*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sqrt[102])^(

2/3)*(8726 + 864*Sqrt[102]))^2*(3*(33*33^(1/3) + 4*3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102

])^(1/3) - 44*(81 + 8*Sqrt[102])^(2/3)) - (81 - 8*Sqrt[102])^(2/3)*(16*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(

1/3) + 3*(4363 + 432*Sqrt[102] + 54*(33*(81 + 8*Sqrt[102]))^(1/3))))^2*(99*33^(1/3) + 3*3^(1/6)*11^(2/3)*(27*S

qrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) - 33*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sqrt[102])^(2/3)*(32*3^(

5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3) + 6*(4363 + 432*Sqrt[102] + 54*(33*(81 + 8*Sqrt[102]))^(1/3))))^2*((8

1 - 8*Sqrt[102])^(1/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3)) - 3*(9 + 4*x^2))) - (155520*(72*(3807*3^(1/6)*11^

(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 - 8*Sqrt[102])^(1/3) - 464*(81 - 8*Sqrt[102])^(2/3)*(4363 + 432*Sqrt[102])

+ 464*(81 + 8*Sqrt[102])^(1/3)*(27*33^(2/3) + 8*3^(1/6)*11^(2/3)*Sqrt[34] - 33*(81 + 8*Sqrt[102])^(1/3))) - 1

1*(8019*3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 - 8*Sqrt[102])^(1/3) - 497*(81 - 8*Sqrt[102])^(2/3)*(43

63 + 432*Sqrt[102]) + 497*(81 + 8*Sqrt[102])^(1/3)*(27*33^(2/3) + 8*3^(1/6)*11^(2/3)*Sqrt[34] - 33*(81 + 8*Sqr

t[102])^(1/3)))*E^4))/((33*33^(1/3) + 3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 33

*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sqrt[102])^(2/3)*(8726 + 864*Sqrt[102]))*(33 - (33*33^(1/3))/(81 + 8*Sqrt[

102])^(2/3) - (33*(81 + 8*Sqrt[102]))^(2/3) - 2*(81 - 8*Sqrt[102])^(2/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3))

^2)*((81 - 8*Sqrt[102])^(2/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3))^2 + 4*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102

])^(2/3) - (33*(81 + 8*Sqrt[102]))^(2/3)))*((81 - 8*Sqrt[102])^(1/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3)) - 3

*(9 + 4*x^2))) - (1046960640*(2*(1034451 - 2129*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqr

t[102]))^(2/3) + (81 - 8*Sqrt[102])^(2/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3))^2)) - 9*(2129*(81 - 8*Sqrt[102

])^(1/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3)) - 43*(81 - 8*Sqrt[102])^(2/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2

/3))^2 + 86*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqrt[102]))^(2/3)))*(9 + 4*x^2)))/((33

- (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqrt[102]))^(2/3) - 2*(81 - 8*Sqrt[102])^(2/3)*(33^(1/3

) + (81 + 8*Sqrt[102])^(2/3))^2)*((81 - 8*Sqrt[102])^(2/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3))^2 + 4*(33 - (

33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqrt[102]))^(2/3)))*((81 - 8*Sqrt[102])^(1/3)*(33^(1/3) +

(81 + 8*Sqrt[102])^(2/3)) - 3*(9 + 4*x^2))^2*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqrt[1

02]))^(2/3) - 3*(81 - 8*Sqrt[102])^(1/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3))*(9 + 4*x^2) - 9*(9 + 4*x^2)^2)^

2) + (169607623680*(81 + 8*Sqrt[102])^(4/3)*(7040*3^(1/6)*11^(2/3)*(248117*Sqrt[3] + 73728*Sqrt[34]) + 638550*

33^(1/3)*(81 - 8*Sqrt[102])^(2/3)*(4363 + 432*Sqrt[102]) + 11*33^(1/3)*(81 + 8*Sqrt[102])^(2/3)*(12182171 + 12

26304*Sqrt[102]) - 2*(81 + 8*Sqrt[102])^(1/3)*(255634488*3^(1/6)*Sqrt[34]*(891 - 88*Sqrt[102])^(2/3) + 8605923

24*(33*(81 - 8*Sqrt[102]))^(2/3) + 1030073*(81 + 8*Sqrt[102])) - 11*(81 - 8*Sqrt[102])^(1/3)*(455908851 + 4515

2936*Sqrt[102] + 1556928*(33*(81 + 8*Sqrt[102]))^(2/3) + 461312*3^(1/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(2/3)) +

2*(549120*3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34]) + 11*33^(1/3)*(81 + 8*Sqrt[102])^(2/3)*(588417 + 58816*S

qrt[102]) - 3*(81 + 8*Sqrt[102])^(1/3)*(1003104*3^(1/6)*Sqrt[34]*(891 - 88*Sqrt[102])^(2/3) + 3376962*(33*(81

- 8*Sqrt[102]))^(2/3) + 817069*(81 + 8*Sqrt[102])) + 33*(81 - 8*Sqrt[102])^(1/3)*(50845995 + 5034440*Sqrt[102]

- 505737*(33*(81 + 8*Sqrt[102]))^(2/3) - 149848*3^(1/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(2/3)))*(9 + 4*x^2)))/(

(3*(33*33^(1/3) + 4*3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) - 44*(81 + 8*Sqrt[102]

)^(2/3)) - (81 - 8*Sqrt[102])^(2/3)*(16*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3) + 3*(4363 + 432*Sqrt[102]

+ 54*(33*(81 + 8*Sqrt[102]))^(1/3))))^2*(99*33^(1/3) + 3*3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sq

rt[102])^(1/3) - 33*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sqrt[102])^(2/3)*(32*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[10

2])^(1/3) + 6*(4363 + 432*Sqrt[102] + 54*(33*(81 + 8*Sqrt[102]))^(1/3))))^2*((81 - 8*Sqrt[102])^(1/3)*(33^(1/3

) + (81 + 8*Sqrt[102])^(2/3)) - 3*(9 + 4*x^2))^2*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sq

rt[102]))^(2/3) - 3*(81 - 8*Sqrt[102])^(1/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3))*(9 + 4*x^2) - 9*(9 + 4*x^2)

^2)) + (25920*(((33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqrt[102]))^(2/3) + (81 - 8*Sqrt[10

2])^(2/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3))^2)*(400896 - 65604*E^4))/6 + 9*(81 - 8*Sqrt[102])^(1/3)*(33^(1

/3) + (81 + 8*Sqrt[102])^(2/3))*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqrt[102]))^(2/3))*

(376 - 121*E^4) + 9*((81 - 8*Sqrt[102])^(1/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3))*(33408 - 5467*E^4) + (66 -

(66*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - 2*(33*(81 + 8*Sqrt[102]))^(2/3) - (81 - 8*Sqrt[102])^(2/3)*(33^(1/3)

+ (81 + 8*Sqrt[102])^(2/3))^2)*(376 - 121*E^4))*(9 + 4*x^2)))/((33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) -

(33*(81 + 8*Sqrt[102]))^(2/3) - 2*(81 - 8*Sqrt[102])^(2/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3))^2)*((81 - 8*

Sqrt[102])^(2/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3))^2 + 4*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (3

3*(81 + 8*Sqrt[102]))^(2/3)))*((81 - 8*Sqrt[102])^(1/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3)) - 3*(9 + 4*x^2))

*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqrt[102]))^(2/3) - 3*(81 - 8*Sqrt[102])^(1/3)*(33

^(1/3) + (81 + 8*Sqrt[102])^(2/3))*(9 + 4*x^2) - 9*(9 + 4*x^2)^2)) + (414596413440*(81 + 8*Sqrt[102])^(1/3)*Sq

rt[3/(44*33^(1/3)*(81 + 8*Sqrt[102])^(2/3) - (81 + 8*Sqrt[102])^(1/3)*(3564 + 352*Sqrt[102] + 8*3^(1/6)*Sqrt[3

4]*(891 - 88*Sqrt[102])^(2/3) + 27*(33*(81 - 8*Sqrt[102]))^(2/3)) + 3^(1/6)*11^(2/3)*(13089*Sqrt[3] + 3888*Sqr

t[34] - 2*(27*Sqrt[3] + 8*Sqrt[34])*(81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[102])^(2/3)))]*(31848325223076427156

4496*3^(5/6)*Sqrt[34]*(891 - 88*Sqrt[102])^(1/3) - 853252141290622554308472*3^(1/6)*Sqrt[34]*(891 - 88*Sqrt[10

2])^(2/3) + 3216523178690219046846285*(33*(81 - 8*Sqrt[102]))^(1/3) - 2872474738210886656356259*(33*(81 - 8*Sq

rt[102]))^(2/3) + 11*(70467083683294018291648*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3) - 362728001039962223

7432*3^(1/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(2/3) - 2*(81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[102])^(2/3)*(852473

89656959752974933 + 8440749341401385431048*Sqrt[102]) + 6*(81 - 8*Sqrt[102])^(2/3)*(81 + 8*Sqrt[102])^(1/3)*(1

44397278229220444881209 + 14297461025118184803248*Sqrt[102]) + 9*(79075851072377972147819*(33*(81 + 8*Sqrt[102

]))^(1/3) - 1356804902883446036645*(33*(81 + 8*Sqrt[102]))^(2/3) + 129551477760*(53750368555299 + 532207953591

2*Sqrt[102]))))*ArcTan[(33^(1/3)*(81 + 8*Sqrt[102] + (81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[102])^(2/3)) + 6*(8

1 + 8*Sqrt[102])^(2/3)*(9 + 4*x^2))/Sqrt[3*(44*33^(1/3)*(81 + 8*Sqrt[102])^(2/3) - (81 + 8*Sqrt[102])^(1/3)*(3

564 + 352*Sqrt[102] + 8*3^(1/6)*Sqrt[34]*(891 - 88*Sqrt[102])^(2/3) + 27*(33*(81 - 8*Sqrt[102]))^(2/3)) + 3^(1

/6)*11^(2/3)*(13089*Sqrt[3] + 3888*Sqrt[34] - 2*(27*Sqrt[3] + 8*Sqrt[34])*(81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqr

t[102])^(2/3)))]])/((66*33^(1/3) - 3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 165*(

81 + 8*Sqrt[102])^(2/3) + 7*(81 - 8*Sqrt[102])^(2/3)*(4363 + 432*Sqrt[102]))*(99*33^(1/3) + 12*3^(1/6)*11^(2/3

)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) - 132*(81 + 8*Sqrt[102])^(2/3) - (81 - 8*Sqrt[102])^(2/3)

*(16*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3) + 3*(4363 + 432*Sqrt[102] + 54*(33*(81 + 8*Sqrt[102]))^(1/3))

))^2*(99*33^(1/3) + 3*3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) - 33*(81 + 8*Sqrt[10

2])^(2/3) + (81 - 8*Sqrt[102])^(2/3)*(32*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3) + 6*(4363 + 432*Sqrt[102]

+ 54*(33*(81 + 8*Sqrt[102]))^(1/3))))^2*(484*(4363*3^(2/3) + 1296*3^(1/6)*Sqrt[34])*(47993 - 4752*Sqrt[102])^

(1/3) + 17452*33^(1/3)*(81 - 8*Sqrt[102])^(2/3)*(38071417 + 3769632*Sqrt[102]) + 33*(9*3^(1/6)*11^(2/3)*(76190

827*Sqrt[3] + 22632048*Sqrt[34]) + 8957*(81 + 8*Sqrt[102])^(1/3)*(705915 + 69896*Sqrt[102]) + 9*33^(1/3)*(81 +

8*Sqrt[102])^(2/3)*(21225443 + 2101632*Sqrt[102])) + 4*(81 - 8*Sqrt[102])^(1/3)*(609911528*3^(1/6)*Sqrt[34]*(

891 + 88*Sqrt[102])^(2/3) + 27*(7538140566 + 746387136*Sqrt[102] + 76046969*(33*(81 + 8*Sqrt[102]))^(2/3)))))

+ (160*Sqrt[3/(44*33^(1/3)*(81 + 8*Sqrt[102])^(2/3) - (81 + 8*Sqrt[102])^(1/3)*(3564 + 352*Sqrt[102] + 8*3^(1/

6)*Sqrt[34]*(891 - 88*Sqrt[102])^(2/3) + 27*(33*(81 - 8*Sqrt[102]))^(2/3)) + 3^(1/6)*11^(2/3)*(13089*Sqrt[3] +

3888*Sqrt[34] - 2*(27*Sqrt[3] + 8*Sqrt[34])*(81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[102])^(2/3)))]*(48*(3*3^(1/

6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34]) + 33^(1/3)*(81 + 8*Sqrt[102])^(5/3)) - (2*33^(1/3)*(81 + 8*Sqrt[102])^(2

/3)*(1447 + 144*Sqrt[102]) - 3^(1/6)*11^(2/3)*(10173*Sqrt[3] + 3024*Sqrt[34] + 2*(27*Sqrt[3] + 8*Sqrt[34])*(81

- 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[102])^(2/3)) - (81 + 8*Sqrt[102])^(1/3)*(8*3^(1/6)*Sqrt[34]*(891 - 88*Sqrt[

102])^(2/3) + 27*(33*(81 - 8*Sqrt[102]))^(2/3) - 22*(81 + 8*Sqrt[102])))*E^4)*ArcTan[(33^(1/3)*(81 + 8*Sqrt[10

2] + (81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[102])^(2/3)) + 6*(81 + 8*Sqrt[102])^(2/3)*(9 + 4*x^2))/Sqrt[3*(44*3

3^(1/3)*(81 + 8*Sqrt[102])^(2/3) - (81 + 8*Sqrt[102])^(1/3)*(3564 + 352*Sqrt[102] + 8*3^(1/6)*Sqrt[34]*(891 -

88*Sqrt[102])^(2/3) + 27*(33*(81 - 8*Sqrt[102]))^(2/3)) + 3^(1/6)*11^(2/3)*(13089*Sqrt[3] + 3888*Sqrt[34] - 2*

(27*Sqrt[3] + 8*Sqrt[34])*(81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[102])^(2/3)))]])/(33*33^(1/3) + 3^(1/6)*11^(2/

3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 33*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sqrt[102])^(2/3)

*(8726 + 864*Sqrt[102])) - (285120*Sqrt[3/(44*33^(1/3)*(81 + 8*Sqrt[102])^(2/3) - (81 + 8*Sqrt[102])^(1/3)*(35

64 + 352*Sqrt[102] + 8*3^(1/6)*Sqrt[34]*(891 - 88*Sqrt[102])^(2/3) + 27*(33*(81 - 8*Sqrt[102]))^(2/3)) + 3^(1/

6)*11^(2/3)*(13089*Sqrt[3] + 3888*Sqrt[34] - 2*(27*Sqrt[3] + 8*Sqrt[34])*(81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt

[102])^(2/3)))]*(24*(88*3^(1/6)*11^(2/3)*(103359383*Sqrt[3] + 30702312*Sqrt[34])*(81 - 8*Sqrt[102])^(1/3) + 42

*3^(1/6)*11^(2/3)*(284498189*Sqrt[3] + 84508032*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 54*33^(1/3)*(52216399515

+ 5170193984*Sqrt[102]) - (81 + 8*Sqrt[102])^(2/3)*(4120795107 + 408019480*Sqrt[102] - 524532576*3^(5/6)*Sqrt[

34]*(891 - 88*Sqrt[102])^(1/3) - 5297500954*(33*(81 - 8*Sqrt[102]))^(1/3)) + (81 - 8*Sqrt[102])^(2/3)*(1110826

2155*(33*(81 + 8*Sqrt[102]))^(1/3) + 1099881840*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3) - 94*(52439871339

+ 5192320976*Sqrt[102]))) - 11*(6*3^(1/6)*11^(2/3)*(951350221*Sqrt[3] + 282593128*Sqrt[34])*(81 - 8*Sqrt[102])

^(1/3) - 24*3^(1/6)*11^(2/3)*(187758093*Sqrt[3] + 55772786*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 117504*33^(1/3

)*(8422987 + 834000*Sqrt[102]) - (81 + 8*Sqrt[102])^(2/3)*(2478352239 + 245393544*Sqrt[102] - 435693184*3^(5/6

)*Sqrt[34]*(891 - 88*Sqrt[102])^(1/3) - 4400279022*(33*(81 - 8*Sqrt[102]))^(1/3)) + (81 - 8*Sqrt[102])^(2/3)*(

6443039985*(33*(81 + 8*Sqrt[102]))^(1/3) + 637956016*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3) - 66*(5243987

1339 + 5192320976*Sqrt[102])))*E^4)*ArcTan[(33^(1/3)*(81 + 8*Sqrt[102] + (81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt

[102])^(2/3)) + 6*(81 + 8*Sqrt[102])^(2/3)*(9 + 4*x^2))/Sqrt[3*(44*33^(1/3)*(81 + 8*Sqrt[102])^(2/3) - (81 + 8

*Sqrt[102])^(1/3)*(3564 + 352*Sqrt[102] + 8*3^(1/6)*Sqrt[34]*(891 - 88*Sqrt[102])^(2/3) + 27*(33*(81 - 8*Sqrt[

102]))^(2/3)) + 3^(1/6)*11^(2/3)*(13089*Sqrt[3] + 3888*Sqrt[34] - 2*(27*Sqrt[3] + 8*Sqrt[34])*(81 - 8*Sqrt[102

])^(1/3)*(81 + 8*Sqrt[102])^(2/3)))]])/((81 + 8*Sqrt[102])^(2/3)*(33*33^(1/3) + 3^(1/6)*11^(2/3)*(27*Sqrt[3] +

8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 33*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sqrt[102])^(2/3)*(8726 + 864*Sqr

t[102]))^2*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqrt[102]))^(2/3) - 2*(81 - 8*Sqrt[102])

^(2/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3))^2)*((81 - 8*Sqrt[102])^(2/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3)

)^2 + 4*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqrt[102]))^(2/3)))) + (226143498240*(81 +

8*Sqrt[102])*(1738285920*3^(1/3)*(47993 - 4752*Sqrt[102])^(2/3)*(384502464 + 38071417*Sqrt[102]) + 11*(16*(81

+ 8*Sqrt[102])^(1/3)*(4481016020965*33^(2/3) + 1331060111766*3^(1/6)*11^(2/3)*Sqrt[34] + 153385263152100*3^(1/

6)*Sqrt[34]*(4363 - 432*Sqrt[102])^(2/3) + 516371738823360*(3*(4363 - 432*Sqrt[102]))^(2/3)) - 57024*33^(1/3)*

(90845842010 + 8995088025*Sqrt[102] - 1910562*(891 - 88*Sqrt[102])^(1/3)*(705915 + 69896*Sqrt[102])^(2/3)) + 3

^(1/6)*11^(2/3)*(81 - 8*Sqrt[102])^(1/3)*(4378135716635999*Sqrt[3] + 1300500091878696*Sqrt[34] + 10787482596*S

qrt[34]*(3*(705915 + 69896*Sqrt[102]))^(2/3)) + 22*(81 + 8*Sqrt[102])^(2/3)*(76075611264*3^(5/6)*Sqrt[34]*(891

- 88*Sqrt[102])^(1/3) + 768922808392*(33*(81 - 8*Sqrt[102]))^(1/3) - 9*(16412724045845 + 1625103072198*Sqrt[1

02])) - (81 - 8*Sqrt[102])^(2/3)*(1181284644860609*(33*(81 + 8*Sqrt[102]))^(1/3) + 116964607354128*3^(5/6)*Sqr

t[34]*(891 + 88*Sqrt[102])^(1/3) + 352*(50657344332912 + 5015823749251*Sqrt[102]))))*Log[(81 - 8*Sqrt[102])^(1

/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3)) - 3*(9 + 4*x^2)])/((33*33^(1/3) + 3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*S

qrt[34])*(81 + 8*Sqrt[102])^(1/3) + 33*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sqrt[102])^(2/3)*(8726 + 864*Sqrt[10

2]))^3*(3*(33*33^(1/3) + 4*3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) - 44*(81 + 8*Sq

rt[102])^(2/3)) - (81 - 8*Sqrt[102])^(2/3)*(16*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3) + 3*(4363 + 432*Sqr

t[102] + 54*(33*(81 + 8*Sqrt[102]))^(1/3))))^2*(99*33^(1/3) + 3*3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81

+ 8*Sqrt[102])^(1/3) - 33*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sqrt[102])^(2/3)*(32*3^(5/6)*Sqrt[34]*(891 + 88*

Sqrt[102])^(1/3) + 6*(4363 + 432*Sqrt[102] + 54*(33*(81 + 8*Sqrt[102]))^(1/3))))^2) - (160*(81 + 8*Sqrt[102])^

(2/3)*(48 - (36 + (33*(81 - 8*Sqrt[102]))^(1/3) + (81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[102])^(2/3))*E^4)*Log[

(81 - 8*Sqrt[102])^(1/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3)) - 3*(9 + 4*x^2)])/(33*33^(1/3) + 3^(1/6)*11^(2/

3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 33*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sqrt[102])^(2/3)

*(8726 + 864*Sqrt[102])) + (77760*(8*(1785420*3^(1/6)*11^(2/3)*(4363*Sqrt[3] + 1296*Sqrt[34])*(81 - 8*Sqrt[102

])^(1/3) + 96679*(81 - 8*Sqrt[102])^(2/3)*(81 + 8*Sqrt[102]) + 11*(3^(1/6)*11^(2/3)*(1164901*Sqrt[3] + 347760*

Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) - 451008*(81 + 8*Sqrt[102])^(5/3) + 43470*33^(1/3)*(4363 + 432*Sqrt[102])))

- 11*(210348*3^(1/6)*11^(2/3)*(4363*Sqrt[3] + 1296*Sqrt[34])*(81 - 8*Sqrt[102])^(1/3) + 187*3^(1/6)*11^(2/3)*

(20615*Sqrt[3] + 6144*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 22627*(81 - 8*Sqrt[102])^(2/3)*(81 + 8*Sqrt[102]) -

590436*(81 + 8*Sqrt[102])^(5/3) + 143616*33^(1/3)*(4363 + 432*Sqrt[102]))*E^4)*Log[(81 - 8*Sqrt[102])^(1/3)*(

33^(1/3) + (81 + 8*Sqrt[102])^(2/3)) - 3*(9 + 4*x^2)])/((81 + 8*Sqrt[102])^(1/3)*(33*33^(1/3) + 3^(1/6)*11^(2/

3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 33*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sqrt[102])^(2/3)

*(8726 + 864*Sqrt[102]))^2*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqrt[102]))^(2/3) - 2*(8

1 - 8*Sqrt[102])^(2/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3))^2)*((81 - 8*Sqrt[102])^(2/3)*(33^(1/3) + (81 + 8*

Sqrt[102])^(2/3))^2 + 4*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqrt[102]))^(2/3)))) + (621

894620160*(81 + 8*Sqrt[102])^(1/3)*(34560*3^(1/6)*11^(2/3)*(1263162156398600025*Sqrt[3] + 375215071661757424*S

qrt[34]) + 440*33^(1/3)*(81 + 8*Sqrt[102])^(2/3)*(1980867501946689645 + 196135109001451318*Sqrt[102]) + 33^(1/

3)*(81 - 8*Sqrt[102])^(2/3)*(281786966869329896033223 + 27901067288851652293088*Sqrt[102]) - 3*(81 + 8*Sqrt[10

2])^(1/3)*(5436518160565005182647 + 538295509903150854888*Sqrt[102] + 178660586364808709344*3^(1/6)*Sqrt[34]*(

891 - 88*Sqrt[102])^(2/3) + 601461158094016696512*(33*(81 - 8*Sqrt[102]))^(2/3)) - (81 - 8*Sqrt[102])^(1/3)*(3

570556233580657116312 + 353537748173990524848*Sqrt[102] - 19453773044641851502*(33*(81 + 8*Sqrt[102]))^(2/3) -

5778631674533398320*3^(1/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(2/3)))*Log[11*33^(1/3) + 3^(1/6)*11^(2/3)*(27*Sqrt

[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) - 11*(81 + 8*Sqrt[102])^(2/3) + 33^(1/3)*(81 + 8*Sqrt[102] + (81 -

8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[102])^(2/3))*(9 + 4*x^2) + 3*(81 + 8*Sqrt[102])^(2/3)*(9 + 4*x^2)^2])/((66*33^

(1/3) - 3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 165*(81 + 8*Sqrt[102])^(2/3) + 7

*(81 - 8*Sqrt[102])^(2/3)*(4363 + 432*Sqrt[102]))*(99*33^(1/3) + 12*3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])

*(81 + 8*Sqrt[102])^(1/3) - 132*(81 + 8*Sqrt[102])^(2/3) - (81 - 8*Sqrt[102])^(2/3)*(16*3^(5/6)*Sqrt[34]*(891

+ 88*Sqrt[102])^(1/3) + 3*(4363 + 432*Sqrt[102] + 54*(33*(81 + 8*Sqrt[102]))^(1/3))))^2*(99*33^(1/3) + 3*3^(1/

6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) - 33*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sqrt[10

2])^(2/3)*(32*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3) + 6*(4363 + 432*Sqrt[102] + 54*(33*(81 + 8*Sqrt[102]

))^(1/3))))^2*(484*(4363*3^(2/3) + 1296*3^(1/6)*Sqrt[34])*(47993 - 4752*Sqrt[102])^(1/3) + 17452*33^(1/3)*(81

- 8*Sqrt[102])^(2/3)*(38071417 + 3769632*Sqrt[102]) + 33*(9*3^(1/6)*11^(2/3)*(76190827*Sqrt[3] + 22632048*Sqrt

[34]) + 8957*(81 + 8*Sqrt[102])^(1/3)*(705915 + 69896*Sqrt[102]) + 9*33^(1/3)*(81 + 8*Sqrt[102])^(2/3)*(212254

43 + 2101632*Sqrt[102])) + 4*(81 - 8*Sqrt[102])^(1/3)*(609911528*3^(1/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(2/3) +

27*(7538140566 + 746387136*Sqrt[102] + 76046969*(33*(81 + 8*Sqrt[102]))^(2/3))))) + (80*(81 + 8*Sqrt[102])^(2

/3)*(48 - (36 + (33*(81 - 8*Sqrt[102]))^(1/3) + (33*(81 + 8*Sqrt[102]))^(1/3))*E^4)*Log[11*33^(1/3) + 3^(1/6)*

11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) - 11*(81 + 8*Sqrt[102])^(2/3) + 33^(1/3)*(81 + 8*S

qrt[102] + (81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[102])^(2/3))*(9 + 4*x^2) + 3*(81 + 8*Sqrt[102])^(2/3)*(9 + 4*

x^2)^2])/(33*33^(1/3) + 3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 33*(81 + 8*Sqrt[

102])^(2/3) + (81 - 8*Sqrt[102])^(2/3)*(8726 + 864*Sqrt[102])) - (38880*(72399028608*3^(1/6)*Sqrt[34]*(891 - 8

8*Sqrt[102])^(2/3) + 243731450192*(33*(81 - 8*Sqrt[102]))^(2/3) + 88*(7755*(81 - 8*Sqrt[102])^(2/3)*(81 + 8*Sq

rt[102])^(4/3) - 168649272*(33*(81 + 8*Sqrt[102]))^(1/3) + 1165935*(33*(81 + 8*Sqrt[102]))^(2/3) - 16698880*3^

(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3) + 347760*3^(1/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(2/3) + 4003236*(4363

+ 432*Sqrt[102]) - 20*(81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[102])^(2/3)*(10329831 + 1022792*Sqrt[102])) - 121

*(2050679808 + 203046912*Sqrt[102] + 192550608*3^(1/6)*Sqrt[34]*(891 - 88*Sqrt[102])^(2/3) + 648221942*(33*(81

- 8*Sqrt[102]))^(2/3) + 147015*(81 - 8*Sqrt[102])^(2/3)*(81 + 8*Sqrt[102])^(1/3) + 14520*Sqrt[102]*(81 - 8*Sq

rt[102])^(2/3)*(81 + 8*Sqrt[102])^(1/3) - 39969128*(81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[102])^(2/3) - 3957472

*Sqrt[102]*(81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[102])^(2/3) - 35279470*(33*(81 + 8*Sqrt[102]))^(1/3) + 350697

*(33*(81 + 8*Sqrt[102]))^(2/3) - 3493216*3^(5/6)*Sqrt[34]*(891 + 88*Sqrt[102])^(1/3) + 104448*3^(1/6)*Sqrt[34]

*(891 + 88*Sqrt[102])^(2/3))*E^4)*Log[11*33^(1/3) + 3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[10

2])^(1/3) - 11*(81 + 8*Sqrt[102])^(2/3) + 33^(1/3)*(81 + 8*Sqrt[102] + (81 - 8*Sqrt[102])^(1/3)*(81 + 8*Sqrt[1

02])^(2/3))*(9 + 4*x^2) + 3*(81 + 8*Sqrt[102])^(2/3)*(9 + 4*x^2)^2])/((81 + 8*Sqrt[102])^(2/3)*(33*33^(1/3) +

3^(1/6)*11^(2/3)*(27*Sqrt[3] + 8*Sqrt[34])*(81 + 8*Sqrt[102])^(1/3) + 33*(81 + 8*Sqrt[102])^(2/3) + (81 - 8*Sq

rt[102])^(2/3)*(8726 + 864*Sqrt[102]))^2*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqrt[102])

)^(2/3) - 2*(81 - 8*Sqrt[102])^(2/3)*(33^(1/3) + (81 + 8*Sqrt[102])^(2/3))^2)*((81 - 8*Sqrt[102])^(2/3)*(33^(1

/3) + (81 + 8*Sqrt[102])^(2/3))^2 + 4*(33 - (33*33^(1/3))/(81 + 8*Sqrt[102])^(2/3) - (33*(81 + 8*Sqrt[102]))^(

2/3))))

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /

; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]

, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +

c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +

b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &

& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp

andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -

4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp

[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x

)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*

c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2

*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -

b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,

c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||

IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F

reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2073

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->

x^2)^p*Q^q, x], x] /; !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,

Rule 2079

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S

qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[(e + f*x)^m*Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/

3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r

/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0

]

Rule 2081

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C

oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2

)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x

] && PolyQ[P3, x, 3]

Rule 2101

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Qn^(p + 1)

)/(n*(p + 1)*Coeff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[P

m, x, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && NeQ[p,

-1]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /

; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {80 x \left (86265+51666 x^2+8801 x^4\right )}{\left (27+58 x^2+27 x^4+4 x^6\right )^3}+\frac {80 x \left (-3 \left (1351-291 e^4\right )-6 \left (310-67 e^4\right ) x^2-\left (288-49 e^4\right ) x^4\right )}{\left (27+58 x^2+27 x^4+4 x^6\right )^2}+\frac {80 x \left (-16+21 e^4+4 e^4 x^2\right )}{27+58 x^2+27 x^4+4 x^6}\right ) \, dx\\ &=-\left (80 \int \frac {x \left (86265+51666 x^2+8801 x^4\right )}{\left (27+58 x^2+27 x^4+4 x^6\right )^3} \, dx\right )+80 \int \frac {x \left (-3 \left (1351-291 e^4\right )-6 \left (310-67 e^4\right ) x^2-\left (288-49 e^4\right ) x^4\right )}{\left (27+58 x^2+27 x^4+4 x^6\right )^2} \, dx+80 \int \frac {x \left (-16+21 e^4+4 e^4 x^2\right )}{27+58 x^2+27 x^4+4 x^6} \, dx\\ &=\frac {44005}{3 \left (27+58 x^2+27 x^4+4 x^6\right )^2}+\frac {10 \left (288-49 e^4\right )}{3 \left (27+58 x^2+27 x^4+4 x^6\right )}-\frac {10}{3} \int \frac {1049444 x+289476 x^3}{\left (27+58 x^2+27 x^4+4 x^6\right )^3} \, dx+\frac {10}{3} \int \frac {-4 \left (15966-3817 e^4\right ) x-36 \left (376-121 e^4\right ) x^3}{\left (27+58 x^2+27 x^4+4 x^6\right )^2} \, dx+40 \operatorname {Subst}\left (\int \frac {-16+21 e^4+4 e^4 x}{27+58 x+27 x^2+4 x^3} \, dx,x,x^2\right )\\ &=\frac {44005}{3 \left (27+58 x^2+27 x^4+4 x^6\right )^2}+\frac {10 \left (288-49 e^4\right )}{3 \left (27+58 x^2+27 x^4+4 x^6\right )}-\frac {10}{3} \int \frac {x \left (1049444+289476 x^2\right )}{\left (27+58 x^2+27 x^4+4 x^6\right )^3} \, dx+\frac {10}{3} \int \frac {x \left (-4 \left (15966-3817 e^4\right )-36 \left (376-121 e^4\right ) x^2\right )}{\left (27+58 x^2+27 x^4+4 x^6\right )^2} \, dx+40 \operatorname {Subst}\left (\int \frac {\frac {1}{12} \left (-108 e^4+12 \left (-16+21 e^4\right )\right )+4 e^4 x}{-\frac {99}{8}-\frac {11 x}{4}+4 x^3} \, dx,x,\frac {1}{4} \left (9+4 x^2\right )\right )\\ &=\frac {44005}{3 \left (27+58 x^2+27 x^4+4 x^6\right )^2}+\frac {10 \left (288-49 e^4\right )}{3 \left (27+58 x^2+27 x^4+4 x^6\right )}-\frac {5}{3} \operatorname {Subst}\left (\int \frac {1049444+289476 x}{\left (27+58 x+27 x^2+4 x^3\right )^3} \, dx,x,x^2\right )+\frac {5}{3} \operatorname {Subst}\left (\int \frac {-4 \left (15966-3817 e^4\right )-36 \left (376-121 e^4\right ) x}{\left (27+58 x+27 x^2+4 x^3\right )^2} \, dx,x,x^2\right )+640 \operatorname {Subst}\left (\int \frac {\frac {1}{12} \left (-108 e^4+12 \left (-16+21 e^4\right )\right )+4 e^4 x}{\left (-\frac {1}{3} \sqrt [3]{81-8 \sqrt {102}} \left (\sqrt [3]{33}+\left (81+8 \sqrt {102}\right )^{2/3}\right )+4 x\right ) \left (\frac {1}{9} \left (-33+\frac {33 \sqrt [3]{33}}{\left (81+8 \sqrt {102}\right )^{2/3}}+\left (33 \left (81+8 \sqrt {102}\right )\right )^{2/3}\right )+\frac {4}{3} \sqrt [3]{81-8 \sqrt {102}} \left (\sqrt [3]{33}+\left (81+8 \sqrt {102}\right )^{2/3}\right ) x+16 x^2\right )} \, dx,x,\frac {1}{4} \left (9+4 x^2\right )\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A] time = 0.05, size = 60, normalized size = 1.88 \begin {gather*} \frac {160 \left (8 \left (3+x^2\right )^4-e^4 \left (3+x^2\right )^2 \left (27+58 x^2+27 x^4+4 x^6\right )\right )}{\left (27+58 x^2+27 x^4+4 x^6\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-16588800*x - 30965760*x^3 - 24883200*x^5 - 10936320*x^7 - 2744320*x^9 - 368640*x^11 - 20480*x^13 +

E^4*(3110400*x + 10414080*x^3 + 12960000*x^5 + 8542720*x^7 + 3310080*x^9 + 760320*x^11 + 96000*x^13 + 5120*x^

15))/(19683 + 126846*x^2 + 331533*x^4 + 457552*x^6 + 369117*x^8 + 184710*x^10 + 58563*x^12 + 11532*x^14 + 1296

*x^16 + 64*x^18),x]

[Out]

(160*(8*(3 + x^2)^4 - E^4*(3 + x^2)^2*(27 + 58*x^2 + 27*x^4 + 4*x^6)))/(27 + 58*x^2 + 27*x^4 + 4*x^6)^2

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fricas [B] time = 0.59, size = 89, normalized size = 2.78 \begin {gather*} \frac {160 \, {\left (8 \, x^{8} + 96 \, x^{6} + 432 \, x^{4} + 864 \, x^{2} - {\left (4 \, x^{10} + 51 \, x^{8} + 256 \, x^{6} + 618 \, x^{4} + 684 \, x^{2} + 243\right )} e^{4} + 648\right )}}{16 \, x^{12} + 216 \, x^{10} + 1193 \, x^{8} + 3348 \, x^{6} + 4822 \, x^{4} + 3132 \, x^{2} + 729} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5120*x^15+96000*x^13+760320*x^11+3310080*x^9+8542720*x^7+12960000*x^5+10414080*x^3+3110400*x)*exp(

4)-20480*x^13-368640*x^11-2744320*x^9-10936320*x^7-24883200*x^5-30965760*x^3-16588800*x)/(64*x^18+1296*x^16+11

532*x^14+58563*x^12+184710*x^10+369117*x^8+457552*x^6+331533*x^4+126846*x^2+19683),x, algorithm="fricas")

[Out]

160*(8*x^8 + 96*x^6 + 432*x^4 + 864*x^2 - (4*x^10 + 51*x^8 + 256*x^6 + 618*x^4 + 684*x^2 + 243)*e^4 + 648)/(16

*x^12 + 216*x^10 + 1193*x^8 + 3348*x^6 + 4822*x^4 + 3132*x^2 + 729)

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giac [B] time = 0.30, size = 82, normalized size = 2.56 \begin {gather*} -\frac {160 \, {\left (4 \, x^{10} e^{4} + 51 \, x^{8} e^{4} - 8 \, x^{8} + 256 \, x^{6} e^{4} - 96 \, x^{6} + 618 \, x^{4} e^{4} - 432 \, x^{4} + 684 \, x^{2} e^{4} - 864 \, x^{2} + 243 \, e^{4} - 648\right )}}{{\left (4 \, x^{6} + 27 \, x^{4} + 58 \, x^{2} + 27\right )}^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5120*x^15+96000*x^13+760320*x^11+3310080*x^9+8542720*x^7+12960000*x^5+10414080*x^3+3110400*x)*exp(

4)-20480*x^13-368640*x^11-2744320*x^9-10936320*x^7-24883200*x^5-30965760*x^3-16588800*x)/(64*x^18+1296*x^16+11

532*x^14+58563*x^12+184710*x^10+369117*x^8+457552*x^6+331533*x^4+126846*x^2+19683),x, algorithm="giac")

[Out]

-160*(4*x^10*e^4 + 51*x^8*e^4 - 8*x^8 + 256*x^6*e^4 - 96*x^6 + 618*x^4*e^4 - 432*x^4 + 684*x^2*e^4 - 864*x^2 +

243*e^4 - 648)/(4*x^6 + 27*x^4 + 58*x^2 + 27)^2

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maple [B] time = 0.12, size = 74, normalized size = 2.31


method	result	size

norman	\(\frac {-640 \,{\mathrm e}^{4} x^{10}+\left (1280-8160 \,{\mathrm e}^{4}\right ) x^{8}+\left (15360-40960 \,{\mathrm e}^{4}\right ) x^{6}+\left (69120-98880 \,{\mathrm e}^{4}\right ) x^{4}+\left (138240-109440 \,{\mathrm e}^{4}\right ) x^{2}+103680-38880 \,{\mathrm e}^{4}}{\left (4 x^{6}+27 x^{4}+58 x^{2}+27\right )^{2}}\)	\(74\)
default	\(\frac {-640 \,{\mathrm e}^{4} x^{10}+10240 \left (-\frac {51 \,{\mathrm e}^{4}}{64}+\frac {1}{8}\right ) x^{8}+10240 \left (-4 \,{\mathrm e}^{4}+\frac {3}{2}\right ) x^{6}+10240 \left (-\frac {309 \,{\mathrm e}^{4}}{32}+\frac {27}{4}\right ) x^{4}+10240 \left (-\frac {171 \,{\mathrm e}^{4}}{16}+\frac {27}{2}\right ) x^{2}-38880 \,{\mathrm e}^{4}+103680}{\left (4 x^{6}+27 x^{4}+58 x^{2}+27\right )^{2}}\)	\(75\)
risch	\(\frac {6480-40 \,{\mathrm e}^{4} x^{10}+\left (80-510 \,{\mathrm e}^{4}\right ) x^{8}+\left (-2560 \,{\mathrm e}^{4}+960\right ) x^{6}+\left (-6180 \,{\mathrm e}^{4}+4320\right ) x^{4}+\left (8640-6840 \,{\mathrm e}^{4}\right ) x^{2}-2430 \,{\mathrm e}^{4}}{x^{12}+\frac {27}{2} x^{10}+\frac {1193}{16} x^{8}+\frac {837}{4} x^{6}+\frac {2411}{8} x^{4}+\frac {783}{4} x^{2}+\frac {729}{16}}\)	\(87\)
gosper	\(-\frac {160 \left (4 x^{8} {\mathrm e}^{4}+39 x^{6} {\mathrm e}^{4}-8 x^{6}+139 x^{4} {\mathrm e}^{4}-72 x^{4}+201 x^{2} {\mathrm e}^{4}-216 x^{2}+81 \,{\mathrm e}^{4}-216\right ) \left (x^{2}+3\right )}{16 x^{12}+216 x^{10}+1193 x^{8}+3348 x^{6}+4822 x^{4}+3132 x^{2}+729}\)	\(91\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((5120*x^15+96000*x^13+760320*x^11+3310080*x^9+8542720*x^7+12960000*x^5+10414080*x^3+3110400*x)*exp(4)-204

80*x^13-368640*x^11-2744320*x^9-10936320*x^7-24883200*x^5-30965760*x^3-16588800*x)/(64*x^18+1296*x^16+11532*x^

14+58563*x^12+184710*x^10+369117*x^8+457552*x^6+331533*x^4+126846*x^2+19683),x,method=_RETURNVERBOSE)

[Out]

(-640*exp(4)*x^10+(1280-8160*exp(4))*x^8+(15360-40960*exp(4))*x^6+(69120-98880*exp(4))*x^4+(138240-109440*exp(

4))*x^2+103680-38880*exp(4))/(4*x^6+27*x^4+58*x^2+27)^2

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maxima [B] time = 0.44, size = 92, normalized size = 2.88 \begin {gather*} -\frac {160 \, {\left (4 \, x^{10} e^{4} + x^{8} {\left (51 \, e^{4} - 8\right )} + 32 \, x^{6} {\left (8 \, e^{4} - 3\right )} + 6 \, x^{4} {\left (103 \, e^{4} - 72\right )} + 36 \, x^{2} {\left (19 \, e^{4} - 24\right )} + 243 \, e^{4} - 648\right )}}{16 \, x^{12} + 216 \, x^{10} + 1193 \, x^{8} + 3348 \, x^{6} + 4822 \, x^{4} + 3132 \, x^{2} + 729} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5120*x^15+96000*x^13+760320*x^11+3310080*x^9+8542720*x^7+12960000*x^5+10414080*x^3+3110400*x)*exp(

4)-20480*x^13-368640*x^11-2744320*x^9-10936320*x^7-24883200*x^5-30965760*x^3-16588800*x)/(64*x^18+1296*x^16+11

532*x^14+58563*x^12+184710*x^10+369117*x^8+457552*x^6+331533*x^4+126846*x^2+19683),x, algorithm="maxima")

[Out]

-160*(4*x^10*e^4 + x^8*(51*e^4 - 8) + 32*x^6*(8*e^4 - 3) + 6*x^4*(103*e^4 - 72) + 36*x^2*(19*e^4 - 24) + 243*e

^4 - 648)/(16*x^12 + 216*x^10 + 1193*x^8 + 3348*x^6 + 4822*x^4 + 3132*x^2 + 729)

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mupad [B] time = 1.29, size = 65, normalized size = 2.03 \begin {gather*} -\frac {160\,{\left (x^2+3\right )}^2\,\left (27\,{\mathrm {e}}^4+58\,x^2\,{\mathrm {e}}^4+27\,x^4\,{\mathrm {e}}^4+4\,x^6\,{\mathrm {e}}^4-48\,x^2-8\,x^4-72\right )}{{\left (4\,x^6+27\,x^4+58\,x^2+27\right )}^2} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(16588800*x - exp(4)*(3110400*x + 10414080*x^3 + 12960000*x^5 + 8542720*x^7 + 3310080*x^9 + 760320*x^11 +

96000*x^13 + 5120*x^15) + 30965760*x^3 + 24883200*x^5 + 10936320*x^7 + 2744320*x^9 + 368640*x^11 + 20480*x^13

)/(126846*x^2 + 331533*x^4 + 457552*x^6 + 369117*x^8 + 184710*x^10 + 58563*x^12 + 11532*x^14 + 1296*x^16 + 64*

x^18 + 19683),x)

[Out]

-(160*(x^2 + 3)^2*(27*exp(4) + 58*x^2*exp(4) + 27*x^4*exp(4) + 4*x^6*exp(4) - 48*x^2 - 8*x^4 - 72))/(58*x^2 +

27*x^4 + 4*x^6 + 27)^2

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sympy [B] time = 8.80, size = 87, normalized size = 2.72 \begin {gather*} \frac {- 640 x^{10} e^{4} + x^{8} \left (1280 - 8160 e^{4}\right ) + x^{6} \left (15360 - 40960 e^{4}\right ) + x^{4} \left (69120 - 98880 e^{4}\right ) + x^{2} \left (138240 - 109440 e^{4}\right ) - 38880 e^{4} + 103680}{16 x^{12} + 216 x^{10} + 1193 x^{8} + 3348 x^{6} + 4822 x^{4} + 3132 x^{2} + 729} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5120*x**15+96000*x**13+760320*x**11+3310080*x**9+8542720*x**7+12960000*x**5+10414080*x**3+3110400*

x)*exp(4)-20480*x**13-368640*x**11-2744320*x**9-10936320*x**7-24883200*x**5-30965760*x**3-16588800*x)/(64*x**1

8+1296*x**16+11532*x**14+58563*x**12+184710*x**10+369117*x**8+457552*x**6+331533*x**4+126846*x**2+19683),x)

[Out]

(-640*x**10*exp(4) + x**8*(1280 - 8160*exp(4)) + x**6*(15360 - 40960*exp(4)) + x**4*(69120 - 98880*exp(4)) + x

**2*(138240 - 109440*exp(4)) - 38880*exp(4) + 103680)/(16*x**12 + 216*x**10 + 1193*x**8 + 3348*x**6 + 4822*x**

4 + 3132*x**2 + 729)

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