3.64 \(\int \frac{\sqrt{3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx\)
Optimal. Leaf size=223 \[ \frac{\sqrt{2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}+\frac{(13665 x+3464) \sqrt{2 x^2-x+3}}{84568 \left (5 x^2+3 x+2\right )}+\frac{\sqrt{\frac{1}{682} \left (112285869463+79399380740 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (112285869463+79399380740 \sqrt{2}\right )}} \left (\left (1235163+872375 \sqrt{2}\right ) x+362788 \sqrt{2}+509587\right )}{\sqrt{2 x^2-x+3}}\right )}{169136}-\frac{\sqrt{\frac{1}{682} \left (79399380740 \sqrt{2}-112285869463\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (79399380740 \sqrt{2}-112285869463\right )}} \left (\left (1235163-872375 \sqrt{2}\right ) x-362788 \sqrt{2}+509587\right )}{\sqrt{2 x^2-x+3}}\right )}{169136} \]
[Out]
((3 + 10*x)*Sqrt[3 - x + 2*x^2])/(62*(2 + 3*x + 5*x^2)^2) + ((3464 + 13665*x)*Sq
rt[3 - x + 2*x^2])/(84568*(2 + 3*x + 5*x^2)) + (Sqrt[(112285869463 + 79399380740
*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(112285869463 + 79399380740*Sqrt[2]))]*(50958
7 + 362788*Sqrt[2] + (1235163 + 872375*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/169136
- (Sqrt[(-112285869463 + 79399380740*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-11228
5869463 + 79399380740*Sqrt[2]))]*(509587 - 362788*Sqrt[2] + (1235163 - 872375*Sq
rt[2])*x))/Sqrt[3 - x + 2*x^2]])/169136
_______________________________________________________________________________________
Rubi [A] time = 0.970659, antiderivative size = 223, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222
\[ \frac{\sqrt{2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}+\frac{(13665 x+3464) \sqrt{2 x^2-x+3}}{84568 \left (5 x^2+3 x+2\right )}+\frac{\sqrt{\frac{1}{682} \left (112285869463+79399380740 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (112285869463+79399380740 \sqrt{2}\right )}} \left (\left (1235163+872375 \sqrt{2}\right ) x+362788 \sqrt{2}+509587\right )}{\sqrt{2 x^2-x+3}}\right )}{169136}-\frac{\sqrt{\frac{1}{682} \left (79399380740 \sqrt{2}-112285869463\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (79399380740 \sqrt{2}-112285869463\right )}} \left (\left (1235163-872375 \sqrt{2}\right ) x-362788 \sqrt{2}+509587\right )}{\sqrt{2 x^2-x+3}}\right )}{169136} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[3 - x + 2*x^2]/(2 + 3*x + 5*x^2)^3,x]
[Out]
((3 + 10*x)*Sqrt[3 - x + 2*x^2])/(62*(2 + 3*x + 5*x^2)^2) + ((3464 + 13665*x)*Sq
rt[3 - x + 2*x^2])/(84568*(2 + 3*x + 5*x^2)) + (Sqrt[(112285869463 + 79399380740
*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(112285869463 + 79399380740*Sqrt[2]))]*(50958
7 + 362788*Sqrt[2] + (1235163 + 872375*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/169136
- (Sqrt[(-112285869463 + 79399380740*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-11228
5869463 + 79399380740*Sqrt[2]))]*(509587 - 362788*Sqrt[2] + (1235163 - 872375*Sq
rt[2])*x))/Sqrt[3 - x + 2*x^2]])/169136
_______________________________________________________________________________________
Rubi in Sympy [A] time = 104.314, size = 252, normalized size = 1.13 \[ \frac{\left (10 x + 3\right ) \sqrt{2 x^{2} - x + 3}}{62 \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{\left (\frac{150315 x}{2} + 19052\right ) \sqrt{2 x^{2} - x + 3}}{465124 \left (5 x^{2} + 3 x + 2\right )} + \frac{\sqrt{682} \left (\frac{61660027}{4} + 10974337 \sqrt{2}\right ) \left (4686088 \sqrt{2} + \frac{13317381}{2}\right ) \operatorname{atan}{\left (\frac{4 \sqrt{341} \left (x \left (\frac{105557375 \sqrt{2}}{4} + \frac{149454723}{4}\right ) + \frac{61660027}{4} + 10974337 \sqrt{2}\right )}{3751 \sqrt{112285869463 + 79399380740 \sqrt{2}} \sqrt{2 x^{2} - x + 3}} \right )}}{211106295004 \sqrt{112285869463 + 79399380740 \sqrt{2}}} + \frac{\sqrt{682} \left (- 10974337 \sqrt{2} + \frac{61660027}{4}\right ) \left (- 4686088 \sqrt{2} + \frac{13317381}{2}\right ) \operatorname{atanh}{\left (\frac{4 \sqrt{341} \left (x \left (- \frac{105557375 \sqrt{2}}{4} + \frac{149454723}{4}\right ) - 10974337 \sqrt{2} + \frac{61660027}{4}\right )}{3751 \sqrt{-112285869463 + 79399380740 \sqrt{2}} \sqrt{2 x^{2} - x + 3}} \right )}}{211106295004 \sqrt{-112285869463 + 79399380740 \sqrt{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*x**2-x+3)**(1/2)/(5*x**2+3*x+2)**3,x)
[Out]
(10*x + 3)*sqrt(2*x**2 - x + 3)/(62*(5*x**2 + 3*x + 2)**2) + (150315*x/2 + 19052
)*sqrt(2*x**2 - x + 3)/(465124*(5*x**2 + 3*x + 2)) + sqrt(682)*(61660027/4 + 109
74337*sqrt(2))*(4686088*sqrt(2) + 13317381/2)*atan(4*sqrt(341)*(x*(105557375*sqr
t(2)/4 + 149454723/4) + 61660027/4 + 10974337*sqrt(2))/(3751*sqrt(112285869463 +
79399380740*sqrt(2))*sqrt(2*x**2 - x + 3)))/(211106295004*sqrt(112285869463 + 7
9399380740*sqrt(2))) + sqrt(682)*(-10974337*sqrt(2) + 61660027/4)*(-4686088*sqrt
(2) + 13317381/2)*atanh(4*sqrt(341)*(x*(-105557375*sqrt(2)/4 + 149454723/4) - 10
974337*sqrt(2) + 61660027/4)/(3751*sqrt(-112285869463 + 79399380740*sqrt(2))*sqr
t(2*x**2 - x + 3)))/(211106295004*sqrt(-112285869463 + 79399380740*sqrt(2)))
_______________________________________________________________________________________
Mathematica [C] time = 6.47419, size = 1170, normalized size = 5.25 \[ \sqrt{2 x^2-x+3} \left (\frac{10 x+3}{62 \left (5 x^2+3 x+2\right )^2}+\frac{13665 x+3464}{84568 \left (5 x^2+3 x+2\right )}\right )-\frac{5 i \left (-174475 i+6521 \sqrt{31}\right ) \tan ^{-1}\left (\frac{31 \left (46775785100 \sqrt{31} x^4+1313174142725 i x^4+254982903010 \sqrt{31} x^3-3744647381480 i x^3+345136479754 \sqrt{31} x^2+3786698475623 i x^2+210477093398 \sqrt{31} x-3659080865574 i x+621237299826 \sqrt{31}+779181710662 i\right )}{-32372991877825 i \sqrt{31} x^4+38797325297500 x^4+1587987614800 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3-13468529326720 i \sqrt{31} x^3+103190181962890 x^3+555795665180 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2-36221356993731 i \sqrt{31} x^2+18159288904922 x^2+396996903700 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x-13553199916122 i \sqrt{31} x+173254405285214 x-158798761480 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}+6160809644426 i \sqrt{31}+31886584896738}\right )}{169136 \sqrt{682 \left (13+i \sqrt{31}\right )}}-\frac{5 i \left (174475 i+6521 \sqrt{31}\right ) \tanh ^{-1}\left (\frac{32372991877825 \sqrt{31} x^4-38797325297500 i x^4+8733931881400 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3+13468529326720 \sqrt{31} x^3-103190181962890 i x^3-19770445804260 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2+36221356993731 \sqrt{31} x^2-18159288904922 i x^2-11512910207300 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x+13553199916122 \sqrt{31} x-173254405285214 i x-10004321973240 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}-6160809644426 \sqrt{31}-31886584896738 i}{1450049338100 \sqrt{31} x^4+153820084388525 i x^4+7904469993310 \sqrt{31} x^3-366664166073320 i x^3+10699230872374 \sqrt{31} x^2+394738353028687 i x^2+6524789895338 \sqrt{31} x+350041661437994 i x+19258356294606 \sqrt{31}+293442889929478 i}\right )}{169136 \sqrt{682 \left (-13+i \sqrt{31}\right )}}+\frac{5 i \left (174475 i+6521 \sqrt{31}\right ) \log \left (\left (-10 i x+\sqrt{31}-3 i\right )^2 \left (10 i x+\sqrt{31}+3 i\right )^2\right )}{338272 \sqrt{682 \left (-13+i \sqrt{31}\right )}}-\frac{5 \left (-174475 i+6521 \sqrt{31}\right ) \log \left (\left (-10 i x+\sqrt{31}-3 i\right )^2 \left (10 i x+\sqrt{31}+3 i\right )^2\right )}{338272 \sqrt{682 \left (13+i \sqrt{31}\right )}}-\frac{5 i \left (174475 i+6521 \sqrt{31}\right ) \log \left (\left (5 x^2+3 x+2\right ) \left (44 \sqrt{31} x^2+327 i x^2-4 i \sqrt{682 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x-22 \sqrt{31} x+469 i x+i \sqrt{682 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}+66 \sqrt{31}-142 i\right )\right )}{338272 \sqrt{682 \left (-13+i \sqrt{31}\right )}}+\frac{5 \left (-174475 i+6521 \sqrt{31}\right ) \log \left (\left (5 x^2+3 x+2\right ) \left (44 \sqrt{31} x^2-817 i x^2+22 i \sqrt{22 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x-22 \sqrt{31} x+1041 i x-63 i \sqrt{22 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}+66 \sqrt{31}-1858 i\right )\right )}{338272 \sqrt{682 \left (13+i \sqrt{31}\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[3 - x + 2*x^2]/(2 + 3*x + 5*x^2)^3,x]
[Out]
Sqrt[3 - x + 2*x^2]*((3 + 10*x)/(62*(2 + 3*x + 5*x^2)^2) + (3464 + 13665*x)/(845
68*(2 + 3*x + 5*x^2))) - (((5*I)/169136)*(-174475*I + 6521*Sqrt[31])*ArcTan[(31*
(779181710662*I + 621237299826*Sqrt[31] - (3659080865574*I)*x + 210477093398*Sqr
t[31]*x + (3786698475623*I)*x^2 + 345136479754*Sqrt[31]*x^2 - (3744647381480*I)*
x^3 + 254982903010*Sqrt[31]*x^3 + (1313174142725*I)*x^4 + 46775785100*Sqrt[31]*x
^4))/(31886584896738 + (6160809644426*I)*Sqrt[31] + 173254405285214*x - (1355319
9916122*I)*Sqrt[31]*x + 18159288904922*x^2 - (36221356993731*I)*Sqrt[31]*x^2 + 1
03190181962890*x^3 - (13468529326720*I)*Sqrt[31]*x^3 + 38797325297500*x^4 - (323
72991877825*I)*Sqrt[31]*x^4 - (158798761480*I)*Sqrt[682*(13 + I*Sqrt[31])]*Sqrt[
3 - x + 2*x^2] + (396996903700*I)*Sqrt[682*(13 + I*Sqrt[31])]*x*Sqrt[3 - x + 2*x
^2] + (555795665180*I)*Sqrt[682*(13 + I*Sqrt[31])]*x^2*Sqrt[3 - x + 2*x^2] + (15
87987614800*I)*Sqrt[682*(13 + I*Sqrt[31])]*x^3*Sqrt[3 - x + 2*x^2])])/Sqrt[682*(
13 + I*Sqrt[31])] - (((5*I)/169136)*(174475*I + 6521*Sqrt[31])*ArcTanh[(-3188658
4896738*I - 6160809644426*Sqrt[31] - (173254405285214*I)*x + 13553199916122*Sqrt
[31]*x - (18159288904922*I)*x^2 + 36221356993731*Sqrt[31]*x^2 - (103190181962890
*I)*x^3 + 13468529326720*Sqrt[31]*x^3 - (38797325297500*I)*x^4 + 32372991877825*
Sqrt[31]*x^4 - 10004321973240*Sqrt[22*(-13 + I*Sqrt[31])]*Sqrt[3 - x + 2*x^2] -
11512910207300*Sqrt[22*(-13 + I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2] - 1977044580426
0*Sqrt[22*(-13 + I*Sqrt[31])]*x^2*Sqrt[3 - x + 2*x^2] + 8733931881400*Sqrt[22*(-
13 + I*Sqrt[31])]*x^3*Sqrt[3 - x + 2*x^2])/(293442889929478*I + 19258356294606*S
qrt[31] + (350041661437994*I)*x + 6524789895338*Sqrt[31]*x + (394738353028687*I)
*x^2 + 10699230872374*Sqrt[31]*x^2 - (366664166073320*I)*x^3 + 7904469993310*Sqr
t[31]*x^3 + (153820084388525*I)*x^4 + 1450049338100*Sqrt[31]*x^4)])/Sqrt[682*(-1
3 + I*Sqrt[31])] - (5*(-174475*I + 6521*Sqrt[31])*Log[(-3*I + Sqrt[31] - (10*I)*
x)^2*(3*I + Sqrt[31] + (10*I)*x)^2])/(338272*Sqrt[682*(13 + I*Sqrt[31])]) + (((5
*I)/338272)*(174475*I + 6521*Sqrt[31])*Log[(-3*I + Sqrt[31] - (10*I)*x)^2*(3*I +
Sqrt[31] + (10*I)*x)^2])/Sqrt[682*(-13 + I*Sqrt[31])] - (((5*I)/338272)*(174475
*I + 6521*Sqrt[31])*Log[(2 + 3*x + 5*x^2)*(-142*I + 66*Sqrt[31] + (469*I)*x - 22
*Sqrt[31]*x + (327*I)*x^2 + 44*Sqrt[31]*x^2 + I*Sqrt[682*(-13 + I*Sqrt[31])]*Sqr
t[3 - x + 2*x^2] - (4*I)*Sqrt[682*(-13 + I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2])])/S
qrt[682*(-13 + I*Sqrt[31])] + (5*(-174475*I + 6521*Sqrt[31])*Log[(2 + 3*x + 5*x^
2)*(-1858*I + 66*Sqrt[31] + (1041*I)*x - 22*Sqrt[31]*x - (817*I)*x^2 + 44*Sqrt[3
1]*x^2 - (63*I)*Sqrt[22*(13 + I*Sqrt[31])]*Sqrt[3 - x + 2*x^2] + (22*I)*Sqrt[22*
(13 + I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2])])/(338272*Sqrt[682*(13 + I*Sqrt[31])])
_______________________________________________________________________________________
Maple [B] time = 0.401, size = 43932, normalized size = 197. \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*x^2-x+3)^(1/2)/(5*x^2+3*x+2)^3,x)
[Out]
result too large to display
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{2 \, x^{2} - x + 3}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x^2 - x + 3)/(5*x^2 + 3*x + 2)^3,x, algorithm="maxima")
[Out]
integrate(sqrt(2*x^2 - x + 3)/(5*x^2 + 3*x + 2)^3, x)
_______________________________________________________________________________________
Fricas [A] time = 0.374415, size = 1646, normalized size = 7.38 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x^2 - x + 3)/(5*x^2 + 3*x + 2)^3,x, algorithm="fricas")
[Out]
1/387269780651700194040320*sqrt(39699690370)*930248^(3/4)*sqrt(31)*(4*sqrt(39699
690370)*930248^(1/4)*sqrt(31)*(10849925378121000*x^3 + 9260349775706200*x^2 - 11
2285869463*sqrt(2)*(68325*x^3 + 58315*x^2 + 51362*x + 11020) + 8156221987135760*
x + 1749962351509600)*sqrt(2*x^2 - x + 3)*sqrt((112285869463*sqrt(2) - 158798761
480)/(17830857002429352685240*sqrt(2) - 25216639804852841803569)) - 14205421276*
sqrt(19849845185)*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(615345200
735*(sqrt(39699690370)*930248^(1/4)*(112285869463*sqrt(2)*(x - 6) - 158798761480
*x + 952792568880)*sqrt((112285869463*sqrt(2) - 158798761480)/(17830857002429352
685240*sqrt(2) - 25216639804852841803569)) + 88*sqrt(19849845185)*sqrt(2*x^2 - x
+ 3)*(362788*sqrt(2) - 509587))/(2*sqrt(39699690370)*sqrt(19849845185)*930248^(
1/4)*sqrt(31)*(112285869463*sqrt(2)*x - 158798761480*x)*sqrt(-sqrt(2)*(23*sqrt(3
9699690370)*sqrt(19849845185)*930248^(1/4)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(1938810
59542971914762680500544591271657*x + 80308164349964553475021076455894155685) - 2
74189223892936468237701577000485427342*x - 1135728951930073612876594240886971159
72)*sqrt((112285869463*sqrt(2) - 158798761480)/(17830857002429352685240*sqrt(2)
- 25216639804852841803569)) + 27978898313001090717344418964781724346768361600*x^
2 + 19849845185*sqrt(2)*(392425208930440881841647027891343640*x^2 - 566299033353
5962546294362498068647*sqrt(2)*(49*x^2 - 151*x + 200) - 120931033772441986036915
7167583528360*x + 1601735546654860742210804195474872000) - 989203436362595257530
3331210194060391812493160*sqrt(2)*(2*x^2 - x + 3) - 1398944915650054535867220948
2390862173384180800*x + 41968347469501636076016628447172586520152542400)/(566299
0333535962546294362498068647*sqrt(2)*x^2 - 8008677733274303711054020977374360*x^
2))*sqrt((112285869463*sqrt(2) - 158798761480)/(17830857002429352685240*sqrt(2)
- 25216639804852841803569)) + 19849845185*sqrt(39699690370)*930248^(1/4)*sqrt(31
)*(112285869463*sqrt(2)*(19*x - 22) - 3017176468120*x + 3493572752560)*sqrt((112
285869463*sqrt(2) - 158798761480)/(17830857002429352685240*sqrt(2) - 25216639804
852841803569)) - 54150377664680*sqrt(19849845185)*sqrt(31)*sqrt(2*x^2 - x + 3)*(
77456*sqrt(2) - 110061))) - 14205421276*sqrt(19849845185)*sqrt(2)*(25*x^4 + 30*x
^3 + 29*x^2 + 12*x + 4)*arctan(-615345200735*(sqrt(39699690370)*930248^(1/4)*(11
2285869463*sqrt(2)*(x - 6) - 158798761480*x + 952792568880)*sqrt((112285869463*s
qrt(2) - 158798761480)/(17830857002429352685240*sqrt(2) - 2521663980485284180356
9)) - 88*sqrt(19849845185)*sqrt(2*x^2 - x + 3)*(362788*sqrt(2) - 509587))/(2*sqr
t(39699690370)*sqrt(19849845185)*930248^(1/4)*sqrt(31)*(112285869463*sqrt(2)*x -
158798761480*x)*sqrt(sqrt(2)*(23*sqrt(39699690370)*sqrt(19849845185)*930248^(1/
4)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(193881059542971914762680500544591271657*x + 803
08164349964553475021076455894155685) - 274189223892936468237701577000485427342*x
- 113572895193007361287659424088697115972)*sqrt((112285869463*sqrt(2) - 1587987
61480)/(17830857002429352685240*sqrt(2) - 25216639804852841803569)) - 2797889831
3001090717344418964781724346768361600*x^2 - 19849845185*sqrt(2)*(392425208930440
881841647027891343640*x^2 - 5662990333535962546294362498068647*sqrt(2)*(49*x^2 -
151*x + 200) - 1209310337724419860369157167583528360*x + 1601735546654860742210
804195474872000) + 9892034363625952575303331210194060391812493160*sqrt(2)*(2*x^2
- x + 3) + 13989449156500545358672209482390862173384180800*x - 4196834746950163
6076016628447172586520152542400)/(5662990333535962546294362498068647*sqrt(2)*x^2
- 8008677733274303711054020977374360*x^2))*sqrt((112285869463*sqrt(2) - 1587987
61480)/(17830857002429352685240*sqrt(2) - 25216639804852841803569)) + 1984984518
5*sqrt(39699690370)*930248^(1/4)*sqrt(31)*(112285869463*sqrt(2)*(19*x - 22) - 30
17176468120*x + 3493572752560)*sqrt((112285869463*sqrt(2) - 158798761480)/(17830
857002429352685240*sqrt(2) - 25216639804852841803569)) + 54150377664680*sqrt(198
49845185)*sqrt(31)*sqrt(2*x^2 - x + 3)*(77456*sqrt(2) - 110061))) + sqrt(1984984
5185)*sqrt(31)*(3969969037000*x^4 + 4763962844400*x^3 + 4605164082920*x^2 - 1122
85869463*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 1905585137760*x + 63519
5045920)*log(-79399380740*sqrt(2)*(23*sqrt(39699690370)*sqrt(19849845185)*930248
^(1/4)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(193881059542971914762680500544591271657*x +
80308164349964553475021076455894155685) - 2741892238929364682377015770004854273
42*x - 113572895193007361287659424088697115972)*sqrt((112285869463*sqrt(2) - 158
798761480)/(17830857002429352685240*sqrt(2) - 25216639804852841803569)) + 279788
98313001090717344418964781724346768361600*x^2 + 19849845185*sqrt(2)*(39242520893
0440881841647027891343640*x^2 - 5662990333535962546294362498068647*sqrt(2)*(49*x
^2 - 151*x + 200) - 1209310337724419860369157167583528360*x + 160173554665486074
2210804195474872000) - 9892034363625952575303331210194060391812493160*sqrt(2)*(2
*x^2 - x + 3) - 13989449156500545358672209482390862173384180800*x + 419683474695
01636076016628447172586520152542400)/(5662990333535962546294362498068647*sqrt(2)
*x^2 - 8008677733274303711054020977374360*x^2)) - sqrt(19849845185)*sqrt(31)*(39
69969037000*x^4 + 4763962844400*x^3 + 4605164082920*x^2 - 112285869463*sqrt(2)*(
25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 1905585137760*x + 635195045920)*log(79399
380740*sqrt(2)*(23*sqrt(39699690370)*sqrt(19849845185)*930248^(1/4)*sqrt(2*x^2 -
x + 3)*(sqrt(2)*(193881059542971914762680500544591271657*x + 803081643499645534
75021076455894155685) - 274189223892936468237701577000485427342*x - 113572895193
007361287659424088697115972)*sqrt((112285869463*sqrt(2) - 158798761480)/(1783085
7002429352685240*sqrt(2) - 25216639804852841803569)) - 2797889831300109071734441
8964781724346768361600*x^2 - 19849845185*sqrt(2)*(392425208930440881841647027891
343640*x^2 - 5662990333535962546294362498068647*sqrt(2)*(49*x^2 - 151*x + 200) -
1209310337724419860369157167583528360*x + 1601735546654860742210804195474872000
) + 9892034363625952575303331210194060391812493160*sqrt(2)*(2*x^2 - x + 3) + 139
89449156500545358672209482390862173384180800*x - 4196834746950163607601662844717
2586520152542400)/(5662990333535962546294362498068647*sqrt(2)*x^2 - 800867773327
4303711054020977374360*x^2)))/((3969969037000*x^4 + 4763962844400*x^3 + 46051640
82920*x^2 - 112285869463*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 1905585
137760*x + 635195045920)*sqrt((112285869463*sqrt(2) - 158798761480)/(17830857002
429352685240*sqrt(2) - 25216639804852841803569)))
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{2 x^{2} - x + 3}}{\left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x**2-x+3)**(1/2)/(5*x**2+3*x+2)**3,x)
[Out]
Integral(sqrt(2*x**2 - x + 3)/(5*x**2 + 3*x + 2)**3, x)
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x^2 - x + 3)/(5*x^2 + 3*x + 2)^3,x, algorithm="giac")
[Out]
Exception raised: RuntimeError