3.2929 \(\int x^4 \sqrt{a+b \sqrt{c x^2}} \, dx\)
Optimal. Leaf size=191 \[ \frac{2 a^4 x^5 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^5 \left (c x^2\right )^{5/2}}-\frac{8 a^3 x^5 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^5 \left (c x^2\right )^{5/2}}+\frac{12 a^2 x^5 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^5 \left (c x^2\right )^{5/2}}+\frac{2 x^5 \left (a+b \sqrt{c x^2}\right )^{11/2}}{11 b^5 \left (c x^2\right )^{5/2}}-\frac{8 a x^5 \left (a+b \sqrt{c x^2}\right )^{9/2}}{9 b^5 \left (c x^2\right )^{5/2}} \]
[Out]
(2*a^4*x^5*(a + b*Sqrt[c*x^2])^(3/2))/(3*b^5*(c*x^2)^(5/2)) - (8*a^3*x^5*(a + b*
Sqrt[c*x^2])^(5/2))/(5*b^5*(c*x^2)^(5/2)) + (12*a^2*x^5*(a + b*Sqrt[c*x^2])^(7/2
))/(7*b^5*(c*x^2)^(5/2)) - (8*a*x^5*(a + b*Sqrt[c*x^2])^(9/2))/(9*b^5*(c*x^2)^(5
/2)) + (2*x^5*(a + b*Sqrt[c*x^2])^(11/2))/(11*b^5*(c*x^2)^(5/2))
_______________________________________________________________________________________
Rubi [A] time = 0.171214, antiderivative size = 191, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095
\[ \frac{2 a^4 x^5 \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b^5 \left (c x^2\right )^{5/2}}-\frac{8 a^3 x^5 \left (a+b \sqrt{c x^2}\right )^{5/2}}{5 b^5 \left (c x^2\right )^{5/2}}+\frac{12 a^2 x^5 \left (a+b \sqrt{c x^2}\right )^{7/2}}{7 b^5 \left (c x^2\right )^{5/2}}+\frac{2 x^5 \left (a+b \sqrt{c x^2}\right )^{11/2}}{11 b^5 \left (c x^2\right )^{5/2}}-\frac{8 a x^5 \left (a+b \sqrt{c x^2}\right )^{9/2}}{9 b^5 \left (c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[x^4*Sqrt[a + b*Sqrt[c*x^2]],x]
[Out]
(2*a^4*x^5*(a + b*Sqrt[c*x^2])^(3/2))/(3*b^5*(c*x^2)^(5/2)) - (8*a^3*x^5*(a + b*
Sqrt[c*x^2])^(5/2))/(5*b^5*(c*x^2)^(5/2)) + (12*a^2*x^5*(a + b*Sqrt[c*x^2])^(7/2
))/(7*b^5*(c*x^2)^(5/2)) - (8*a*x^5*(a + b*Sqrt[c*x^2])^(9/2))/(9*b^5*(c*x^2)^(5
/2)) + (2*x^5*(a + b*Sqrt[c*x^2])^(11/2))/(11*b^5*(c*x^2)^(5/2))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 20.0885, size = 180, normalized size = 0.94 \[ \frac{2 a^{4} x^{5} \left (a + b \sqrt{c x^{2}}\right )^{\frac{3}{2}}}{3 b^{5} \left (c x^{2}\right )^{\frac{5}{2}}} - \frac{8 a^{3} x^{5} \left (a + b \sqrt{c x^{2}}\right )^{\frac{5}{2}}}{5 b^{5} \left (c x^{2}\right )^{\frac{5}{2}}} + \frac{12 a^{2} x^{5} \left (a + b \sqrt{c x^{2}}\right )^{\frac{7}{2}}}{7 b^{5} \left (c x^{2}\right )^{\frac{5}{2}}} - \frac{8 a x^{5} \left (a + b \sqrt{c x^{2}}\right )^{\frac{9}{2}}}{9 b^{5} \left (c x^{2}\right )^{\frac{5}{2}}} + \frac{2 x^{5} \left (a + b \sqrt{c x^{2}}\right )^{\frac{11}{2}}}{11 b^{5} \left (c x^{2}\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(a+b*(c*x**2)**(1/2))**(1/2),x)
[Out]
2*a**4*x**5*(a + b*sqrt(c*x**2))**(3/2)/(3*b**5*(c*x**2)**(5/2)) - 8*a**3*x**5*(
a + b*sqrt(c*x**2))**(5/2)/(5*b**5*(c*x**2)**(5/2)) + 12*a**2*x**5*(a + b*sqrt(c
*x**2))**(7/2)/(7*b**5*(c*x**2)**(5/2)) - 8*a*x**5*(a + b*sqrt(c*x**2))**(9/2)/(
9*b**5*(c*x**2)**(5/2)) + 2*x**5*(a + b*sqrt(c*x**2))**(11/2)/(11*b**5*(c*x**2)*
*(5/2))
_______________________________________________________________________________________
Mathematica [A] time = 0.258565, size = 0, normalized size = 0. \[ \int x^4 \sqrt{a+b \sqrt{c x^2}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[x^4*Sqrt[a + b*Sqrt[c*x^2]],x]
[Out]
Integrate[x^4*Sqrt[a + b*Sqrt[c*x^2]], x]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 84, normalized size = 0.4 \[ -{\frac{2\,{x}^{5}}{3465\,{b}^{5}} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}} \left ( -315\,{c}^{2}{x}^{4}{b}^{4}+280\, \left ( c{x}^{2} \right ) ^{3/2}a{b}^{3}-240\,c{x}^{2}{a}^{2}{b}^{2}+192\,\sqrt{c{x}^{2}}{a}^{3}b-128\,{a}^{4} \right ) \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(a+b*(c*x^2)^(1/2))^(1/2),x)
[Out]
-2/3465*x^5*(a+b*(c*x^2)^(1/2))^(3/2)*(-315*c^2*x^4*b^4+280*(c*x^2)^(3/2)*a*b^3-
240*c*x^2*a^2*b^2+192*(c*x^2)^(1/2)*a^3*b-128*a^4)/(c*x^2)^(5/2)/b^5
_______________________________________________________________________________________
Maxima [A] time = 2.03099, size = 2959, normalized size = 15.49 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a)*x^4,x, algorithm="maxima")
[Out]
((253*c^33 + 2558956*c^32 + 7217549950*c^31 + 8987703765844*c^30 + 6036468373437
617*c^29 + 2446429529849811272*c^28 + 642455910258816305144*c^27 + 1147773662812
26527056208*c^26 + 14444206931227366367330858*c^25 + 131365425653725890097887892
0*c^24 + 88007787535651613090646185140*c^23 + 4405711003982878865632262198872*c^
22 + 166544000020720524719921573991514*c^21 + 4789438716064434805459841864162048
*c^20 + 105284116366548048830595983583302024*c^19 + 1773444928146150427905082217
087812880*c^18 + 22894839259775871001829906064713305625*c^17 + 22607666002341147
3110523953150238987500*c^16 + 1700246465246927686150050738273824218750*c^15 + 96
72993246548251837557896244481445312500*c^14 + 4123018572079203526143742593788403
3203125*c^13 + 129956781520382049850939376902099609375000*c^12 + 297694072785916
684263677284641113281250000*c^11 + 484329529502415188750357304687500000000000*c^
10 + 542693518652974490238804687500000000000000*c^9 + 40155973753395455088875000
0000000000000000*c^8 + 184849853908622316875000000000000000000000*c^7 + 48394254
985190280000000000000000000000000*c^6 + 6212164126680000000000000000000000000000
*c^5 + 292206528000000000000000000000000000000*c^4 + 209952000000000000000000000
0000000000*c^3 + (c^33 + 31444*c^32 + 153361414*c^31 + 277761034468*c^30 + 24953
1421449205*c^29 + 128781547874762192*c^28 + 41710765820505500216*c^27 + 89888688
27121079441936*c^26 + 1342840780494748947766706*c^25 + 1432661664245642574279178
48*c^24 + 11159995340528093004218780644*c^23 + 645308253745684073542574566840*c^
22 + 28040775985068292094392289155714*c^21 + 923839137257607053430155106920656*c
^20 + 23217713909073497257593845317682600*c^19 + 4466994343467654515785860342182
89616*c^18 + 6587962278317998652016627783149754125*c^17 + 7441133104745009071492
6280659005912500*c^16 + 641687702611309496545321307620461343750*c^15 + 420162704
0402268957195316129626601562500*c^14 + 20719469703534769824847871522785791015625
*c^13 + 76087449461581872206557314710546875000000*c^12 + 20496011645616511158441
8106022949218750000*c^11 + 396983461479743662087124789843750000000000*c^10 + 538
387172682023504940143750000000000000000*c^9 + 4932589180053796347758750000000000
00000000*c^8 + 290460146783055534875000000000000000000000*c^7 + 1022449625624296
20000000000000000000000000*c^6 + 19203795434390400000000000000000000000000*c^5 +
1572810307200000000000000000000000000000*c^4 + 35411904000000000000000000000000
000000*c^3)*sqrt(c))*b^5*x^5 + (c^33 + 30448*c^32 + 143612758*c^31 + 25118863879
6*c^30 + 217599624606565*c^29 + 108117268374716764*c^28 + 33654923995101723352*c
^27 + 6957523970007441794992*c^26 + 995051698889961908261746*c^25 + 101410165881
894182490782352*c^24 + 7527940968489364320155782596*c^23 + 413724159098907450302
482347816*c^22 + 17037518514719295836702957925282*c^21 + 53026343341023368793771
6137759112*c^20 + 12544173979379497042757936065180200*c^19 + 2262697525506452089
40329076927964720*c^18 + 3114498606543720283441564145645938125*c^17 + 3266395171
3046131242671682730487700000*c^16 + 260004289926401703985972418707308593750*c^15
+ 1560709818236985476370671875848242187500*c^14 + 69988538091335300965470365584
31884765625*c^13 + 23148367524550212705560195893920898437500*c^12 + 555068707674
40315469623732717285156250000*c^11 + 94317102615121972546395374755859375000000*c
^10 + 110142696514314310681040527343750000000000*c^9 + 8476798762251064471730468
7500000000000000*c^8 + 40508998607407646796875000000000000000000*c^7 + 109895246
46462701250000000000000000000000*c^6 + 1459169837032500000000000000000000000000*
c^5 + 70871193000000000000000000000000000000*c^4 + 52488000000000000000000000000
0000000*c^3 + (249*c^32 + 2437164*c^31 + 6643098918*c^30 + 7982949210660*c^29 +
5166069875011357*c^28 + 2013960456350944216*c^27 + 507836214278409411736*c^26 +
86947270401196759876240*c^25 + 10464000135667518734283874*c^24 + 908013593009682
171015749512*c^23 + 57896023661694155810023054756*c^22 + 27508143675872490644223
32807608*c^21 + 98393925961843341373109742290386*c^20 + 266838498242350005370897
7313125600*c^19 + 55107420449030060659564239322581224*c^18 + 8683659179435695921
43765909375954000*c^17 + 10436844833600989868063649482129553125*c^16 + 954208531
71226948139837222228288187500*c^15 + 660229305541320870206161063444589843750*c^1
4 + 3430153973600309932075208741088476562500*c^13 + 1323477048425791487524927970
4156494140625*c^12 + 37363311422181199028698593326416015625000*c^11 + 7566658971
6155422385182353771972656250000*c^10 + 10706111904192729856477580566406250000000
0*c^9 + 102122732595717247514642578125000000000000*c^8 + 62487787043911972019531
250000000000000000*c^7 + 22813859478991729687500000000000000000000*c^6 + 4436156
399339475000000000000000000000000*c^5 + 375484778550000000000000000000000000000*
c^4 + 8721756000000000000000000000000000000*c^3)*sqrt(c))*a*b^4*x^4 - 4*(c^32 +
29710*c^31 + 136568656*c^30 + 232488459946*c^29 + 195743173114099*c^28 + 9438074
7307709584*c^27 + 28462469351818276960*c^26 + 5690177551338578052424*c^25 + 7854
21485648418831104842*c^24 + 77086958845727395767874308*c^23 + 549768281907186436
9019402832*c^22 + 289515233485471762193587809036*c^21 + 113907125133267945031782
49783782*c^20 + 337598068144644814346991158941992*c^19 + 75774016454108002107539
24556281328*c^18 + 129144106012252228858421679966753000*c^17 + 16716978068232815
66736061537218853125*c^16 + 16398697473652695739105288119068718750*c^15 + 121330
007675595121218408345094062500000*c^14 + 671991970693378956717863791361035156250
*c^13 + 2756319624573010910782184457415771484375*c^12 + 825093269293356627685201
6898193359375000*c^11 + 17675330737298814875196104821777343750000*c^10 + 2639531
0102345039267613256835937500000000*c^9 + 265171303096377683952324218750000000000
00*c^8 + 17053962622098817730468750000000000000000*c^7 + 65313010745610903125000
00000000000000000*c^6 + 1329655880537325000000000000000000000000*c^5 + 117603563
850000000000000000000000000000*c^4 + 2848932000000000000000000000000000000*c^3 +
2*(123*c^31 + 1174017*c^30 + 3116696475*c^29 + 3642741915411*c^28 + 22894201778
34530*c^27 + 865409107213907732*c^26 + 211224403111477290428*c^25 + 349383688735
90512859484*c^24 + 4053867839361131120484674*c^23 + 338376358236249991856063294*
c^22 + 20701487602239281351482423130*c^21 + 941134333565416888920784690250*c^20
+ 32110894210931478931787496469520*c^19 + 827795388994782805334001918149812*c^18
+ 16187607756398830013651232826868620*c^17 + 2404667999534064527842504347378475
00*c^16 + 2710875706565572583927732435236496875*c^15 + 2311238037513443046126067
8935541015625*c^14 + 148119641257267753275468014081201171875*c^13 + 707089030760
086530960808683502685546875*c^12 + 2482905805269441071451363165954589843750*c^11
+ 6305256671690250099071271315917968750000*c^10 + 11320298752129488879797019653
320312500000*c^9 + 13937594367446090380968017578125000000000*c^8 + 1128567083340
1971164472656250000000000000*c^7 + 5662949588807759414062500000000000000000*c^6
+ 1609978127654229375000000000000000000000*c^5 + 2235943788637500000000000000000
00000000*c^4 + 11337043500000000000000000000000000000*c^3 + 87480000000000000000
000000000000000*c^2)*sqrt(c))*a^2*b^3*x^3 + 12*(c^31 + 29222*c^30 + 131989476*c^
29 + 220549631950*c^28 + 182054403980255*c^27 + 85951284212292484*c^26 + 2534463
8059811815968*c^25 + 4946658491131916154584*c^24 + 665454644118584444285242*c^23
+ 63533306064757209063076580*c^22 + 4398310610385893237847455976*c^21 + 2243025
25518058209739047940420*c^20 + 8523385281137359786451302784462*c^19 + 2432480324
25468337765646384201760*c^18 + 5239212219133542340480502420489120*c^17 + 8535052
3863191078165738758341235000*c^16 + 1051232702462420068262014831632403125*c^15 +
9760125457240085676442417704652343750*c^14 + 6792098800401774207913530017050781
2500*c^13 + 351197357680378911932532935718261718750*c^12 + 133275293225418043466
9081466278076171875*c^11 + 3650321200872523729722890099487304687500*c^10 + 70555
88854027909397802579956054687500000*c^9 + 93364705099387213396354980468750000000
00*c^8 + 8112634879608292229902343750000000000000*c^7 + 436181880692410199218750
0000000000000000*c^6 + 1326777947026460625000000000000000000000*c^5 + 1968551580
26250000000000000000000000000*c^4 + 10646680500000000000000000000000000000*c^3 +
87480000000000000000000000000000000*c^2 + 2*(122*c^30 + 1144795*c^29 + 29847069
99*c^28 + 3422192283461*c^27 + 2107365773854275*c^26 + 779457823001615248*c^25 +
185879765051665474460*c^24 + 29991710382458596704900*c^23 + 3388413195242546676
199432*c^22 + 274843052171492782792986714*c^21 + 16303176991853388113634967154*c
^20 + 716831808047358679181736749830*c^19 + 23587508929794119145336193685058*c^1
8 + 584547356569314467568355533948052*c^17 + 10948395537265287673170730406379500
*c^16 + 155116276090215374618511676396612500*c^15 + 1659643004103152515665717603
604093750*c^14 + 13352254917894344784818261230888671875*c^13 + 80198653253250011
196332713910693359375*c^12 + 355891673079707619028275747784423828125*c^11 + 1150
152873015260636782281699676513671875*c^10 + 265493547081772636934838121643066406
2500*c^9 + 4264709898101579481994439697265625000000*c^8 + 4601123857507369041332
519531250000000000*c^7 + 3173035953793678934570312500000000000000*c^6 + 13011307
81883657421875000000000000000000*c^5 + 283200180627768750000000000000000000000*c
^4 + 26739220837500000000000000000000000000*c^3 + 690363000000000000000000000000
000000*c^2)*sqrt(c))*a^3*b^2*x^2 - 24*(c^30 + 28979*c^29 + 129728865*c^28 + 2147
09946817*c^27 + 175424729360150*c^26 + 81911977393944084*c^25 + 2386763439120252
9556*c^24 + 4598766595419787735220*c^23 + 610069989949087038610662*c^22 + 573665
49664221202749288378*c^21 + 3905991055707128875010770926*c^20 + 1956021625900585
62386788777038*c^19 + 7285323827632700990474618061840*c^18 + 2033583383935128004
65448614893484*c^17 + 4273475844388426205809239967486500*c^16 + 6772720863304892
9025206537495962500*c^15 + 808727358915038248050198016335140625*c^14 + 724956680
7948818893161180513779296875*c^13 + 48466044976177871402659958222509765625*c^12
+ 239266096150056760942527466119384765625*c^11 + 8602356822448219575550574368286
13281250*c^10 + 2210251137086824413713384136962890625000*c^9 + 39559690494792810
72819201660156250000000*c^8 + 4763019763214843448465820312500000000000*c^7 + 367
3406927808397595703125000000000000000*c^6 + 168915382714514171875000000000000000
0000*c^5 + 413670210404287500000000000000000000000*c^4 + 44125007175000000000000
000000000000000*c^3 + 1293246000000000000000000000000000000*c^2 + (243*c^29 + 22
60611*c^28 + 5839685133*c^27 + 6629674620105*c^26 + 4039306818348400*c^25 + 1477
003668609286412*c^24 + 347891895712128419364*c^23 + 55384654169497405674580*c^22
+ 6166756400536006313788202*c^21 + 492319554678764362836685050*c^20 + 287003629
27999647352259163382*c^19 + 1238061453504658795976684722622*c^18 + 3988969403195
5537300197769308276*c^17 + 965736374745116134671262453002620*c^16 + 176233152301
42149140532220845272500*c^15 + 242505343547381820211816815297262500*c^14 + 25105
58649291266783281237190873046875*c^13 + 19454943027839870676475341947998046875*c
^12 + 111931261530322150990005469598876953125*c^11 + 472517250009358477114024029
449462890625*c^10 + 1440070063785699316009505962524414062500*c^9 + 3099619804548
628324983378295898437500000*c^8 + 4573450746723877891169677734375000000000*c^7 +
4439227951799894634199218750000000000000*c^6 + 26726649797789602734375000000000
00000000*c^5 + 913107736622173125000000000000000000000*c^4 + 1527301508512500000
00000000000000000000*c^3 + 9353434500000000000000000000000000000*c^2 + 874800000
00000000000000000000000000*c)*sqrt(c))*a^4*b*x + 24*(c^29 + 28979*c^28 + 1297288
65*c^27 + 214709946817*c^26 + 175424729360150*c^25 + 81911977393944084*c^24 + 23
867634391202529556*c^23 + 4598766595419787735220*c^22 + 610069989949087038610662
*c^21 + 57366549664221202749288378*c^20 + 3905991055707128875010770926*c^19 + 19
5602162590058562386788777038*c^18 + 7285323827632700990474618061840*c^17 + 20335
8338393512800465448614893484*c^16 + 4273475844388426205809239967486500*c^15 + 67
727208633048929025206537495962500*c^14 + 808727358915038248050198016335140625*c^
13 + 7249566807948818893161180513779296875*c^12 + 484660449761778714026599582225
09765625*c^11 + 239266096150056760942527466119384765625*c^10 + 86023568224482195
7555057436828613281250*c^9 + 2210251137086824413713384136962890625000*c^8 + 3955
969049479281072819201660156250000000*c^7 + 4763019763214843448465820312500000000
000*c^6 + 3673406927808397595703125000000000000000*c^5 + 16891538271451417187500
00000000000000000*c^4 + 413670210404287500000000000000000000000*c^3 + 4412500717
5000000000000000000000000000*c^2 + (243*c^28 + 2260611*c^27 + 5839685133*c^26 +
6629674620105*c^25 + 4039306818348400*c^24 + 1477003668609286412*c^23 + 34789189
5712128419364*c^22 + 55384654169497405674580*c^21 + 6166756400536006313788202*c^
20 + 492319554678764362836685050*c^19 + 28700362927999647352259163382*c^18 + 123
8061453504658795976684722622*c^17 + 39889694031955537300197769308276*c^16 + 9657
36374745116134671262453002620*c^15 + 17623315230142149140532220845272500*c^14 +
242505343547381820211816815297262500*c^13 + 251055864929126678328123719087304687
5*c^12 + 19454943027839870676475341947998046875*c^11 + 1119312615303221509900054
69598876953125*c^10 + 472517250009358477114024029449462890625*c^9 + 144007006378
5699316009505962524414062500*c^8 + 3099619804548628324983378295898437500000*c^7
+ 4573450746723877891169677734375000000000*c^6 + 4439227951799894634199218750000
000000000*c^5 + 2672664979778960273437500000000000000000*c^4 + 91310773662217312
5000000000000000000000*c^3 + 152730150851250000000000000000000000000*c^2 + 93534
34500000000000000000000000000000*c + 87480000000000000000000000000000000)*sqrt(c
) + 1293246000000000000000000000000000000*c)*a^5)*sqrt(b*sqrt(c)*x + a)/((c^34 +
32709*c^33 + 166156194*c^32 + 313848784218*c^31 + 294469940278425*c^30 + 158963
889741950277*c^29 + 53942913469754556576*c^28 + 12201148378415160967656*c^27 + 1
916727611900881583047746*c^26 + 215487201080701089264572138*c^25 + 1772826662321
4387509113175244*c^24 + 1085347191423942138995805492540*c^23 + 50069331004982686
422553600150074*c^22 + 1756559137361209677029762976878226*c^21 + 471649074893956
71284893054638492840*c^20 + 973120016179505695731565952134799736*c^19 + 15455186
919048750791542038868588818525*c^18 + 188885527346329445724075810982572440625*c^
17 + 1772071002728366862097941073371656281250*c^16 + 127028593666369073879455698
20995722656250*c^15 + 69084435936276029012637352745193017578125*c^14 + 282238378
065542048513744444399967041015625*c^13 + 854744024058075360839114990533447265625
000*c^12 + 1885453825409327083405511213049316406250000*c^11 + 296003482019409944
8691930273437500000000000*c^10 + 3206726511270252085969898437500000000000000*c^9
+ 2298258834452828289318750000000000000000000*c^8 + 102649423210554120437500000
0000000000000000*c^7 + 261175070360341800000000000000000000000000*c^6 + 32633630
940600000000000000000000000000000*c^5 + 1496444544000000000000000000000000000000
*c^4 + 10497600000000000000000000000000000000*c^3 + 2*(129*c^33 + 1358088*c^32 +
3992178510*c^31 + 5188254469092*c^30 + 3642062740341821*c^29 + 1545168634611811
116*c^28 + 425504869680671903112*c^27 + 79860855208415962132944*c^26 + 105792054
16850555553082194*c^25 + 1014992544330040094059234080*c^24 + 7190388211914603905
5870044180*c^23 + 3816126136355649616672567516536*c^22 + 15337393997303099259594
1509885042*c^21 + 4704317201176235036305308699382664*c^20 + 11068634295595776755
9282605085857512*c^19 + 2003471049939988842899006194089630480*c^18 + 27917325325
682932130956522490231038125*c^17 + 299066657630330963342577678222634275000*c^16
+ 2454342489151737584438328638188065468750*c^15 + 153405642242797983117672384463
07226562500*c^14 + 72413767119232942192838391775906494140625*c^13 + 255197014414
145705441862975227416992187500*c^12 + 661247327533371121092883907377929687500000
*c^11 + 1234623418450566749592990626953125000000000*c^10 + 161731469103154600746
9761718750000000000000*c^9 + 1433927163780426362384062500000000000000000*c^8 + 8
18575293911949995625000000000000000000000*c^7 + 27980953389866919000000000000000
0000000000*c^6 + 51115570649316000000000000000000000000000*c^5 + 407812903200000
0000000000000000000000000*c^4 + 89579520000000000000000000000000000000*c^3)*sqrt
(c))*b^5)
_______________________________________________________________________________________
Fricas [A] time = 0.207801, size = 132, normalized size = 0.69 \[ \frac{2 \,{\left (315 \, b^{5} c^{3} x^{6} - 40 \, a^{2} b^{3} c^{2} x^{4} - 64 \, a^{4} b c x^{2} +{\left (35 \, a b^{4} c^{2} x^{4} + 48 \, a^{3} b^{2} c x^{2} + 128 \, a^{5}\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b + a}}{3465 \, b^{5} c^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a)*x^4,x, algorithm="fricas")
[Out]
2/3465*(315*b^5*c^3*x^6 - 40*a^2*b^3*c^2*x^4 - 64*a^4*b*c*x^2 + (35*a*b^4*c^2*x^
4 + 48*a^3*b^2*c*x^2 + 128*a^5)*sqrt(c*x^2))*sqrt(sqrt(c*x^2)*b + a)/(b^5*c^3*x)
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{4} \sqrt{a + b \sqrt{c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(a+b*(c*x**2)**(1/2))**(1/2),x)
[Out]
Integral(x**4*sqrt(a + b*sqrt(c*x**2)), x)
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.217329, size = 147, normalized size = 0.77 \[ \frac{2 \,{\left (315 \,{\left (b \sqrt{c} x + a\right )}^{\frac{11}{2}} b^{40} c^{20} - 1540 \,{\left (b \sqrt{c} x + a\right )}^{\frac{9}{2}} a b^{40} c^{20} + 2970 \,{\left (b \sqrt{c} x + a\right )}^{\frac{7}{2}} a^{2} b^{40} c^{20} - 2772 \,{\left (b \sqrt{c} x + a\right )}^{\frac{5}{2}} a^{3} b^{40} c^{20} + 1155 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}} a^{4} b^{40} c^{20}\right )}}{3465 \, b^{45} c^{\frac{45}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(sqrt(c*x^2)*b + a)*x^4,x, algorithm="giac")
[Out]
2/3465*(315*(b*sqrt(c)*x + a)^(11/2)*b^40*c^20 - 1540*(b*sqrt(c)*x + a)^(9/2)*a*
b^40*c^20 + 2970*(b*sqrt(c)*x + a)^(7/2)*a^2*b^40*c^20 - 2772*(b*sqrt(c)*x + a)^
(5/2)*a^3*b^40*c^20 + 1155*(b*sqrt(c)*x + a)^(3/2)*a^4*b^40*c^20)/(b^45*c^(45/2)
)