I just build the pre-release 1.3.8 from github on Linux and got this:
setSimplifyDenomsFlag(true)
r=integrate(x/((x^2 - ((1 + 2*sqrt(-1))/5))*sqrt(x^3 - x)), x);
unparse(r::InputForm)
"((4^(1/4))^3*((-2)*(-1)^(1/2)+11)^(1/4)*log(((((75669959187507629394531250*x
^18+(-669388100504875183105468750)*x^16+1261942088603973388671875000*x^14+959
26225185394287109375000*x^12+(-1562852412462234497070312500)*x^10+86302682757
3776245117187500*x^8+(-33326447010040283203125000)*x^6+(-41164457798004150390
625000)*x^4+3458932042121887207031250*x^2+31664967536926269531250)*(-1)^(1/2)
+((-52386894822120666503906250)*x^18+29103830456733703613281250*x^16+11501833
79650115966796875000*x^14+(-2308748662471771240234375000)*x^12+10368414223194
12231445312500*x^10+378187745809555053710937500*x^8+(-28655678033828735351562
5000)*x^6+28498470783233642578125000*x^4+2047047019004821777343750*x^2+(-5774
1999626159667968750)))*(4^(1/4))^2*(((-2)*(-1)^(1/2)+11)^(1/4))^2+((488944351
673126220703125000*x^17+3352761268615722656250000000*x^15+(-13653188943862915
039062500000)*x^13+12353062629699707031250000000*x^11+(-882893800735473632812
50000)*x^9+(-3137230873107910156250000000)*x^7+748246908187866210937500000*x^
5+7510185241699218750000000*x^3+(-5058944225311279296875000)*x)*(-1)^(1/2)+(1
676380634307861328125000000*x^17+(-6798654794692993164062500000)*x^15+4079192
876815795898437500000*x^13+8519738912582397460937500000*x^11+(-95494091510772
70507812500000)*x^9+1992255449295043945312500000*x^7+412732362747192382812500
000*x^5+(-104635953903198242187500000)*x^3+1281499862670898437500000*x)))*(x^
3+(-1)*x)^(1/2)+(((128056854009628295898437500*x^19+(-50058588385581970214843
7500)*x^17+9313225746154785156250000*x^15+1659616827964782714843750000*x^13+(
-1897476613521575927734375000)*x^11+522278249263763427734375000*x^9+139370560
646057128906250000*x^7+(-64030289649963378906250000)*x^5+33043324947357177734
37500*x^3+152736902236938476562500*x)*(-1)^(1/2)+(23283064365386962890625000*
x^19+(-535510480403900146484375000)*x^17+1732259988784790039062500000*x^15+(-
1598149538040161132812500000)*x^13+(-192597508430480957031250000)*x^11+860676
169395446777343750000*x^9+(-301808118820190429687500000)*x^7+6645917892456054
687500000*x^5+5342066287994384765625000*x^3+(-141561031341552734375000)*x))*4
^(1/4)*(((-2)*(-1)^(1/2)+11)^(1/4))^3+((5820766091346740722656250*x^20+(-3492
45965480804443359375000)*x^18+2005835995078086853027343750*x^16+(-32074749469
75708007812500000)*x^14+960472971200942993164062500*x^12+14509409666061401367
18750000*x^10+(-1016844063997268676757812500)*x^8+131219625473022460937500000
*x^6+19775703549385070800781250*x^4+(-2406537532806396484375000)*x^2+(-186264
5149230957031250))*(-1)^(1/2)+((-32014213502407073974609375)*x^20+46566128730
7739257812500000*x^18+(-554719008505344390869140625)*x^16+(-16838312149047851
56250000000)*x^14+3611529245972633361816406250*x^12+(-19629001617431640625000
00000)*x^10+(-26958063244819641113281250)*x^8+226378440856933593750000000*x^6
+(-32397918403148651123046875)*x^4+(-268220901489257812500000)*x^2+1024454832
0770263671875))*(4^(1/4))^3*((-2)*(-1)^(1/2)+11)^(1/4)))/(1600000*x^20+(-3200
000)*x^18+4160000*x^16+(-3584000)*x^14+2380800*x^12+(-1193984)*x^10+476160*x^
8+(-143360)*x^6+33280*x^4+(-5120)*x^2+512))+((-1)*(4^(1/4))^3*((-2)*(-1)^(1/2
)+11)^(1/4)*log(((((75669959187507629394531250*x^18+(-66938810050487518310546
8750)*x^16+1261942088603973388671875000*x^14+95926225185394287109375000*x^12+
(-1562852412462234497070312500)*x^10+863026827573776245117187500*x^8+(-333264
47010040283203125000)*x^6+(-41164457798004150390625000)*x^4+34589320421218872
07031250*x^2+31664967536926269531250)*(-1)^(1/2)+((-5238689482212066650390625
0)*x^18+29103830456733703613281250*x^16+1150183379650115966796875000*x^14+(-2
308748662471771240234375000)*x^12+1036841422319412231445312500*x^10+378187745
809555053710937500*x^8+(-286556780338287353515625000)*x^6+2849847078323364257
8125000*x^4+2047047019004821777343750*x^2+(-57741999626159667968750)))*(4^(1/
4))^2*(((-2)*(-1)^(1/2)+11)^(1/4))^2+((488944351673126220703125000*x^17+33527
61268615722656250000000*x^15+(-13653188943862915039062500000)*x^13+1235306262
9699707031250000000*x^11+(-88289380073547363281250000)*x^9+(-3137230873107910
156250000000)*x^7+748246908187866210937500000*x^5+7510185241699218750000000*x
^3+(-5058944225311279296875000)*x)*(-1)^(1/2)+(1676380634307861328125000000*x
^17+(-6798654794692993164062500000)*x^15+4079192876815795898437500000*x^13+85
19738912582397460937500000*x^11+(-9549409151077270507812500000)*x^9+199225544
9295043945312500000*x^7+412732362747192382812500000*x^5+(-1046359539031982421
87500000)*x^3+1281499862670898437500000*x)))*(x^3+(-1)*x)^(1/2)+((((-12805685
4009628295898437500)*x^19+500585883855819702148437500*x^17+(-9313225746154785
156250000)*x^15+(-1659616827964782714843750000)*x^13+189747661352157592773437
5000*x^11+(-522278249263763427734375000)*x^9+(-139370560646057128906250000)*x
^7+64030289649963378906250000*x^5+(-3304332494735717773437500)*x^3+(-15273690
2236938476562500)*x)*(-1)^(1/2)+((-23283064365386962890625000)*x^19+535510480
403900146484375000*x^17+(-1732259988784790039062500000)*x^15+1598149538040161
132812500000*x^13+192597508430480957031250000*x^11+(-860676169395446777343750
000)*x^9+301808118820190429687500000*x^7+(-6645917892456054687500000)*x^5+(-5
342066287994384765625000)*x^3+141561031341552734375000*x))*4^(1/4)*(((-2)*(-1
)^(1/2)+11)^(1/4))^3+(((-5820766091346740722656250)*x^20+34924596548080444335
9375000*x^18+(-2005835995078086853027343750)*x^16+320747494697570800781250000
0*x^14+(-960472971200942993164062500)*x^12+(-1450940966606140136718750000)*x^
10+1016844063997268676757812500*x^8+(-131219625473022460937500000)*x^6+(-1977
5703549385070800781250)*x^4+2406537532806396484375000*x^2+1862645149230957031
250)*(-1)^(1/2)+(32014213502407073974609375*x^20+(-46566128730773925781250000
0)*x^18+554719008505344390869140625*x^16+1683831214904785156250000000*x^14+(-
3611529245972633361816406250)*x^12+1962900161743164062500000000*x^10+26958063
244819641113281250*x^8+(-226378440856933593750000000)*x^6+3239791840314865112
3046875*x^4+268220901489257812500000*x^2+(-10244548320770263671875)))*(4^(1/4
))^3*((-2)*(-1)^(1/2)+11)^(1/4)))/(1600000*x^20+(-3200000)*x^18+4160000*x^16+
(-3584000)*x^14+2380800*x^12+(-1193984)*x^10+476160*x^8+(-143360)*x^6+33280*x
^4+(-5120)*x^2+512))+((-1)*(-1)^(1/2)*(4^(1/4))^3*((-2)*(-1)^(1/2)+11)^(1/4)*
log((((((-75669959187507629394531250)*x^18+669388100504875183105468750*x^16+(
-1261942088603973388671875000)*x^14+(-95926225185394287109375000)*x^12+156285
2412462234497070312500*x^10+(-863026827573776245117187500)*x^8+33326447010040
283203125000*x^6+41164457798004150390625000*x^4+(-3458932042121887207031250)*
x^2+(-31664967536926269531250))*(-1)^(1/2)+(52386894822120666503906250*x^18+(
-29103830456733703613281250)*x^16+(-1150183379650115966796875000)*x^14+230874
8662471771240234375000*x^12+(-1036841422319412231445312500)*x^10+(-3781877458
09555053710937500)*x^8+286556780338287353515625000*x^6+(-28498470783233642578
125000)*x^4+(-2047047019004821777343750)*x^2+57741999626159667968750))*(4^(1/
4))^2*(((-2)*(-1)^(1/2)+11)^(1/4))^2+((488944351673126220703125000*x^17+33527
61268615722656250000000*x^15+(-13653188943862915039062500000)*x^13+1235306262
9699707031250000000*x^11+(-88289380073547363281250000)*x^9+(-3137230873107910
156250000000)*x^7+748246908187866210937500000*x^5+7510185241699218750000000*x
^3+(-5058944225311279296875000)*x)*(-1)^(1/2)+(1676380634307861328125000000*x
^17+(-6798654794692993164062500000)*x^15+4079192876815795898437500000*x^13+85
19738912582397460937500000*x^11+(-9549409151077270507812500000)*x^9+199225544
9295043945312500000*x^7+412732362747192382812500000*x^5+(-1046359539031982421
87500000)*x^3+1281499862670898437500000*x)))*(x^3+(-1)*x)^(1/2)+(((2328306436
5386962890625000*x^19+(-535510480403900146484375000)*x^17+1732259988784790039
062500000*x^15+(-1598149538040161132812500000)*x^13+(-19259750843048095703125
0000)*x^11+860676169395446777343750000*x^9+(-301808118820190429687500000)*x^7
+6645917892456054687500000*x^5+5342066287994384765625000*x^3+(-14156103134155
2734375000)*x)*(-1)^(1/2)+((-128056854009628295898437500)*x^19+50058588385581
9702148437500*x^17+(-9313225746154785156250000)*x^15+(-1659616827964782714843
750000)*x^13+1897476613521575927734375000*x^11+(-522278249263763427734375000)
*x^9+(-139370560646057128906250000)*x^7+64030289649963378906250000*x^5+(-3304
332494735717773437500)*x^3+(-152736902236938476562500)*x))*4^(1/4)*(((-2)*(-1
)^(1/2)+11)^(1/4))^3+((32014213502407073974609375*x^20+(-46566128730773925781
2500000)*x^18+554719008505344390869140625*x^16+1683831214904785156250000000*x
^14+(-3611529245972633361816406250)*x^12+1962900161743164062500000000*x^10+26
958063244819641113281250*x^8+(-226378440856933593750000000)*x^6+3239791840314
8651123046875*x^4+268220901489257812500000*x^2+(-10244548320770263671875))*(-
1)^(1/2)+(5820766091346740722656250*x^20+(-349245965480804443359375000)*x^18+
2005835995078086853027343750*x^16+(-3207474946975708007812500000)*x^14+960472
971200942993164062500*x^12+1450940966606140136718750000*x^10+(-10168440639972
68676757812500)*x^8+131219625473022460937500000*x^6+1977570354938507080078125
0*x^4+(-2406537532806396484375000)*x^2+(-1862645149230957031250)))*(4^(1/4))^
3*((-2)*(-1)^(1/2)+11)^(1/4)))/(1600000*x^20+(-3200000)*x^18+4160000*x^16+(-3
584000)*x^14+2380800*x^12+(-1193984)*x^10+476160*x^8+(-143360)*x^6+33280*x^4+
(-5120)*x^2+512))+(-1)^(1/2)*(4^(1/4))^3*((-2)*(-1)^(1/2)+11)^(1/4)*log((((((
-75669959187507629394531250)*x^18+669388100504875183105468750*x^16+(-12619420
88603973388671875000)*x^14+(-95926225185394287109375000)*x^12+156285241246223
4497070312500*x^10+(-863026827573776245117187500)*x^8+33326447010040283203125
000*x^6+41164457798004150390625000*x^4+(-3458932042121887207031250)*x^2+(-316
64967536926269531250))*(-1)^(1/2)+(52386894822120666503906250*x^18+(-29103830
456733703613281250)*x^16+(-1150183379650115966796875000)*x^14+230874866247177
1240234375000*x^12+(-1036841422319412231445312500)*x^10+(-3781877458095550537
10937500)*x^8+286556780338287353515625000*x^6+(-28498470783233642578125000)*x
^4+(-2047047019004821777343750)*x^2+57741999626159667968750))*(4^(1/4))^2*(((
-2)*(-1)^(1/2)+11)^(1/4))^2+((488944351673126220703125000*x^17+33527612686157
22656250000000*x^15+(-13653188943862915039062500000)*x^13+1235306262969970703
1250000000*x^11+(-88289380073547363281250000)*x^9+(-3137230873107910156250000
000)*x^7+748246908187866210937500000*x^5+7510185241699218750000000*x^3+(-5058
944225311279296875000)*x)*(-1)^(1/2)+(1676380634307861328125000000*x^17+(-679
8654794692993164062500000)*x^15+4079192876815795898437500000*x^13+85197389125
82397460937500000*x^11+(-9549409151077270507812500000)*x^9+199225544929504394
5312500000*x^7+412732362747192382812500000*x^5+(-104635953903198242187500000)
*x^3+1281499862670898437500000*x)))*(x^3+(-1)*x)^(1/2)+((((-23283064365386962
890625000)*x^19+535510480403900146484375000*x^17+(-17322599887847900390625000
00)*x^15+1598149538040161132812500000*x^13+192597508430480957031250000*x^11+(
-860676169395446777343750000)*x^9+301808118820190429687500000*x^7+(-664591789
2456054687500000)*x^5+(-5342066287994384765625000)*x^3+1415610313415527343750
00*x)*(-1)^(1/2)+(128056854009628295898437500*x^19+(-500585883855819702148437
500)*x^17+9313225746154785156250000*x^15+1659616827964782714843750000*x^13+(-
1897476613521575927734375000)*x^11+522278249263763427734375000*x^9+1393705606
46057128906250000*x^7+(-64030289649963378906250000)*x^5+330433249473571777343
7500*x^3+152736902236938476562500*x))*4^(1/4)*(((-2)*(-1)^(1/2)+11)^(1/4))^3+
(((-32014213502407073974609375)*x^20+465661287307739257812500000*x^18+(-55471
9008505344390869140625)*x^16+(-1683831214904785156250000000)*x^14+36115292459
72633361816406250*x^12+(-1962900161743164062500000000)*x^10+(-269580632448196
41113281250)*x^8+226378440856933593750000000*x^6+(-32397918403148651123046875
)*x^4+(-268220901489257812500000)*x^2+10244548320770263671875)*(-1)^(1/2)+((-
5820766091346740722656250)*x^20+349245965480804443359375000*x^18+(-2005835995
078086853027343750)*x^16+3207474946975708007812500000*x^14+(-9604729712009429
93164062500)*x^12+(-1450940966606140136718750000)*x^10+1016844063997268676757
812500*x^8+(-131219625473022460937500000)*x^6+(-19775703549385070800781250)*x
^4+2406537532806396484375000*x^2+1862645149230957031250))*(4^(1/4))^3*((-2)*(
-1)^(1/2)+11)^(1/4)))/(1600000*x^20+(-3200000)*x^18+4160000*x^16+(-3584000)*x
^14+2380800*x^12+(-1193984)*x^10+476160*x^8+(-143360)*x^6+33280*x^4+(-5120)*x
^2+512)))))/80"
--Nasser