3.85 \(\int \frac{1}{\sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^3} \, dx\)
Optimal. Leaf size=223 \[ \frac{\sqrt{2 x^2-x+3} (65 x+4)}{1364 \left (5 x^2+3 x+2\right )^2}+\frac{(86265 x+26794) \sqrt{2 x^2-x+3}}{1860496 \left (5 x^2+3 x+2\right )}+\frac{25 \sqrt{\frac{1}{682} \left (6414867847+4536374600 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (6414867847+4536374600 \sqrt{2}\right )}} \left (\left (294669+208915 \sqrt{2}\right ) x+85754 \sqrt{2}+123161\right )}{\sqrt{2 x^2-x+3}}\right )}{3720992}-\frac{25 \sqrt{\frac{1}{682} \left (4536374600 \sqrt{2}-6414867847\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (4536374600 \sqrt{2}-6414867847\right )}} \left (\left (294669-208915 \sqrt{2}\right ) x-85754 \sqrt{2}+123161\right )}{\sqrt{2 x^2-x+3}}\right )}{3720992} \]
[Out]
((4 + 65*x)*Sqrt[3 - x + 2*x^2])/(1364*(2 + 3*x + 5*x^2)^2) + ((26794 + 86265*x)
*Sqrt[3 - x + 2*x^2])/(1860496*(2 + 3*x + 5*x^2)) + (25*Sqrt[(6414867847 + 45363
74600*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(6414867847 + 4536374600*Sqrt[2]))]*(123
161 + 85754*Sqrt[2] + (294669 + 208915*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/372099
2 - (25*Sqrt[(-6414867847 + 4536374600*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-6414
867847 + 4536374600*Sqrt[2]))]*(123161 - 85754*Sqrt[2] + (294669 - 208915*Sqrt[2
])*x))/Sqrt[3 - x + 2*x^2]])/3720992
_______________________________________________________________________________________
Rubi [A] time = 0.968858, 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
\[ \frac{\sqrt{2 x^2-x+3} (65 x+4)}{1364 \left (5 x^2+3 x+2\right )^2}+\frac{(86265 x+26794) \sqrt{2 x^2-x+3}}{1860496 \left (5 x^2+3 x+2\right )}+\frac{25 \sqrt{\frac{1}{682} \left (6414867847+4536374600 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (6414867847+4536374600 \sqrt{2}\right )}} \left (\left (294669+208915 \sqrt{2}\right ) x+85754 \sqrt{2}+123161\right )}{\sqrt{2 x^2-x+3}}\right )}{3720992}-\frac{25 \sqrt{\frac{1}{682} \left (4536374600 \sqrt{2}-6414867847\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (4536374600 \sqrt{2}-6414867847\right )}} \left (\left (294669-208915 \sqrt{2}\right ) x-85754 \sqrt{2}+123161\right )}{\sqrt{2 x^2-x+3}}\right )}{3720992} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)^3),x]
[Out]
((4 + 65*x)*Sqrt[3 - x + 2*x^2])/(1364*(2 + 3*x + 5*x^2)^2) + ((26794 + 86265*x)
*Sqrt[3 - x + 2*x^2])/(1860496*(2 + 3*x + 5*x^2)) + (25*Sqrt[(6414867847 + 45363
74600*Sqrt[2])/682]*ArcTan[(Sqrt[11/(31*(6414867847 + 4536374600*Sqrt[2]))]*(123
161 + 85754*Sqrt[2] + (294669 + 208915*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/372099
2 - (25*Sqrt[(-6414867847 + 4536374600*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-6414
867847 + 4536374600*Sqrt[2]))]*(123161 - 85754*Sqrt[2] + (294669 - 208915*Sqrt[2
])*x))/Sqrt[3 - x + 2*x^2]])/3720992
_______________________________________________________________________________________
Rubi in Sympy [A] time = 104.21, size = 258, normalized size = 1.16 \[ \frac{\left (715 x + 44\right ) \sqrt{2 x^{2} - x + 3}}{15004 \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{\left (\frac{10438065 x}{2} + 1621037\right ) \sqrt{2 x^{2} - x + 3}}{112560008 \left (5 x^{2} + 3 x + 2\right )} + \frac{\sqrt{682} \left (\frac{868577325}{2} + 310422475 \sqrt{2}\right ) \left (\frac{1426732175 \sqrt{2}}{2} + \frac{4098182275}{4}\right ) \operatorname{atan}{\left (\frac{4 \sqrt{341} \left (x \left (\frac{9805110975}{4} + \frac{6951646625 \sqrt{2}}{4}\right ) + \frac{1426732175 \sqrt{2}}{2} + \frac{4098182275}{4}\right )}{1031525 \sqrt{6414867847 + 4536374600 \sqrt{2}} \sqrt{2 x^{2} - x + 3}} \right )}}{14049123932516200 \sqrt{6414867847 + 4536374600 \sqrt{2}}} + \frac{\sqrt{682} \left (- \frac{1426732175 \sqrt{2}}{2} + \frac{4098182275}{4}\right ) \left (- 310422475 \sqrt{2} + \frac{868577325}{2}\right ) \operatorname{atanh}{\left (\frac{4 \sqrt{341} \left (x \left (- \frac{6951646625 \sqrt{2}}{4} + \frac{9805110975}{4}\right ) - \frac{1426732175 \sqrt{2}}{2} + \frac{4098182275}{4}\right )}{1031525 \sqrt{-6414867847 + 4536374600 \sqrt{2}} \sqrt{2 x^{2} - x + 3}} \right )}}{14049123932516200 \sqrt{-6414867847 + 4536374600 \sqrt{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(5*x**2+3*x+2)**3/(2*x**2-x+3)**(1/2),x)
[Out]
(715*x + 44)*sqrt(2*x**2 - x + 3)/(15004*(5*x**2 + 3*x + 2)**2) + (10438065*x/2
+ 1621037)*sqrt(2*x**2 - x + 3)/(112560008*(5*x**2 + 3*x + 2)) + sqrt(682)*(8685
77325/2 + 310422475*sqrt(2))*(1426732175*sqrt(2)/2 + 4098182275/4)*atan(4*sqrt(3
41)*(x*(9805110975/4 + 6951646625*sqrt(2)/4) + 1426732175*sqrt(2)/2 + 4098182275
/4)/(1031525*sqrt(6414867847 + 4536374600*sqrt(2))*sqrt(2*x**2 - x + 3)))/(14049
123932516200*sqrt(6414867847 + 4536374600*sqrt(2))) + sqrt(682)*(-1426732175*sqr
t(2)/2 + 4098182275/4)*(-310422475*sqrt(2) + 868577325/2)*atanh(4*sqrt(341)*(x*(
-6951646625*sqrt(2)/4 + 9805110975/4) - 1426732175*sqrt(2)/2 + 4098182275/4)/(10
31525*sqrt(-6414867847 + 4536374600*sqrt(2))*sqrt(2*x**2 - x + 3)))/(14049123932
516200*sqrt(-6414867847 + 4536374600*sqrt(2)))
_______________________________________________________________________________________
Mathematica [C] time = 6.46702, size = 1170, normalized size = 5.25 \[ \sqrt{2 x^2-x+3} \left (\frac{65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}+\frac{86265 x+26794}{1860496 \left (5 x^2+3 x+2\right )}\right )-\frac{125 i \left (-41783 i+1489 \sqrt{31}\right ) \tan ^{-1}\left (\frac{31 \left (97553324 \sqrt{31} x^4+3245899757 i x^4+557246338 \sqrt{31} x^3-8456927744 i x^3+784505986 \sqrt{31} x^2+8927431079 i x^2+438440750 \sqrt{31} x-8257920150 i x+1411781250 \sqrt{31}+1733669734 i\right )}{-72669503461 i \sqrt{31} x^4+84861105868 x^4+3629099680 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3-29645645200 i \sqrt{31} x^3+237240959890 x^3+1270184888 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2-81049798431 i \sqrt{31} x^2+37412913890 x^2+907274920 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x-31523713098 i \sqrt{31} x+394528763486 x-362909968 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}+14089391258 i \sqrt{31}+74935517250}\right )}{3720992 \sqrt{682 \left (13+i \sqrt{31}\right )}}-\frac{125 i \left (41783 i+1489 \sqrt{31}\right ) \tanh ^{-1}\left (\frac{72669503461 \sqrt{31} x^4-84861105868 i x^4+19960048240 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3+29645645200 \sqrt{31} x^3-237240959890 i x^3-45182291016 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2+81049798431 \sqrt{31} x^2-37412913890 i x^2-26310972680 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x+31523713098 \sqrt{31} x-394528763486 i x-22863327984 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}-14089391258 \sqrt{31}-74935517250 i}{3024153044 \sqrt{31} x^4+343941818333 i x^4+17274636478 \sqrt{31} x^3-841081542656 i x^3+24319685566 \sqrt{31} x^2+893634283351 i x^2+13591663250 \sqrt{31} x+796731376970 i x+43765218750 \sqrt{31}+672076174246 i}\right )}{3720992 \sqrt{682 \left (-13+i \sqrt{31}\right )}}+\frac{125 i \left (41783 i+1489 \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 )}{7441984 \sqrt{682 \left (-13+i \sqrt{31}\right )}}-\frac{125 \left (-41783 i+1489 \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 )}{7441984 \sqrt{682 \left (13+i \sqrt{31}\right )}}-\frac{125 i \left (41783 i+1489 \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 )}{7441984 \sqrt{682 \left (-13+i \sqrt{31}\right )}}+\frac{125 \left (-41783 i+1489 \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 )}{7441984 \sqrt{682 \left (13+i \sqrt{31}\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[3 - x + 2*x^2]*(2 + 3*x + 5*x^2)^3),x]
[Out]
Sqrt[3 - x + 2*x^2]*((4 + 65*x)/(1364*(2 + 3*x + 5*x^2)^2) + (26794 + 86265*x)/(
1860496*(2 + 3*x + 5*x^2))) - (((125*I)/3720992)*(-41783*I + 1489*Sqrt[31])*ArcT
an[(31*(1733669734*I + 1411781250*Sqrt[31] - (8257920150*I)*x + 438440750*Sqrt[3
1]*x + (8927431079*I)*x^2 + 784505986*Sqrt[31]*x^2 - (8456927744*I)*x^3 + 557246
338*Sqrt[31]*x^3 + (3245899757*I)*x^4 + 97553324*Sqrt[31]*x^4))/(74935517250 + (
14089391258*I)*Sqrt[31] + 394528763486*x - (31523713098*I)*Sqrt[31]*x + 37412913
890*x^2 - (81049798431*I)*Sqrt[31]*x^2 + 237240959890*x^3 - (29645645200*I)*Sqrt
[31]*x^3 + 84861105868*x^4 - (72669503461*I)*Sqrt[31]*x^4 - (362909968*I)*Sqrt[6
82*(13 + I*Sqrt[31])]*Sqrt[3 - x + 2*x^2] + (907274920*I)*Sqrt[682*(13 + I*Sqrt[
31])]*x*Sqrt[3 - x + 2*x^2] + (1270184888*I)*Sqrt[682*(13 + I*Sqrt[31])]*x^2*Sqr
t[3 - x + 2*x^2] + (3629099680*I)*Sqrt[682*(13 + I*Sqrt[31])]*x^3*Sqrt[3 - x + 2
*x^2])])/Sqrt[682*(13 + I*Sqrt[31])] - (((125*I)/3720992)*(41783*I + 1489*Sqrt[3
1])*ArcTanh[(-74935517250*I - 14089391258*Sqrt[31] - (394528763486*I)*x + 315237
13098*Sqrt[31]*x - (37412913890*I)*x^2 + 81049798431*Sqrt[31]*x^2 - (23724095989
0*I)*x^3 + 29645645200*Sqrt[31]*x^3 - (84861105868*I)*x^4 + 72669503461*Sqrt[31]
*x^4 - 22863327984*Sqrt[22*(-13 + I*Sqrt[31])]*Sqrt[3 - x + 2*x^2] - 26310972680
*Sqrt[22*(-13 + I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2] - 45182291016*Sqrt[22*(-13 +
I*Sqrt[31])]*x^2*Sqrt[3 - x + 2*x^2] + 19960048240*Sqrt[22*(-13 + I*Sqrt[31])]*x
^3*Sqrt[3 - x + 2*x^2])/(672076174246*I + 43765218750*Sqrt[31] + (796731376970*I
)*x + 13591663250*Sqrt[31]*x + (893634283351*I)*x^2 + 24319685566*Sqrt[31]*x^2 -
(841081542656*I)*x^3 + 17274636478*Sqrt[31]*x^3 + (343941818333*I)*x^4 + 302415
3044*Sqrt[31]*x^4)])/Sqrt[682*(-13 + I*Sqrt[31])] - (125*(-41783*I + 1489*Sqrt[3
1])*Log[(-3*I + Sqrt[31] - (10*I)*x)^2*(3*I + Sqrt[31] + (10*I)*x)^2])/(7441984*
Sqrt[682*(13 + I*Sqrt[31])]) + (((125*I)/7441984)*(41783*I + 1489*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])] - (((125*I)/7441984)*(41783*I + 1489*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])]*Sqrt[3 - x + 2*x^2] - (4*I)*Sqrt[682*(-13 +
I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2])])/Sqrt[682*(-13 + I*Sqrt[31])] + (125*(-4178
3*I + 1489*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[31]*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]
)])/(7441984*Sqrt[682*(13 + I*Sqrt[31])])
_______________________________________________________________________________________
Maple [B] time = 0.037, size = 13040, normalized size = 58.5 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(5*x^2+3*x+2)^3/(2*x^2-x+3)^(1/2),x)
[Out]
result too large to display
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3} \sqrt{2 \, x^{2} - x + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x^2 + 3*x + 2)^3*sqrt(2*x^2 - x + 3)),x, algorithm="maxima")
[Out]
integrate(1/((5*x^2 + 3*x + 2)^3*sqrt(2*x^2 - x + 3)), x)
_______________________________________________________________________________________
Fricas [A] time = 0.37231, size = 1601, normalized size = 7.18 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x^2 + 3*x + 2)^3*sqrt(2*x^2 - x + 3)),x, algorithm="fricas")
[Out]
1/2433873989840816267008*sqrt(22681873)*232562^(3/4)*sqrt(31)*(8*sqrt(22681873)*
232562^(1/4)*sqrt(31)*(3913303548690000*x^3 + 3563458339538000*x^2 - 6414867847*
sqrt(2)*(431325*x^3 + 392765*x^2 + 341572*x + 59044) + 3098997089742400*x + 5356
91403764800)*sqrt(2*x^2 - x + 3)*sqrt((6414867847*sqrt(2) - 9072749200)/(5820048
7126974972400*sqrt(2) - 82307918517524735409)) + 46113488900*sqrt(22681873)*sqrt
(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(31*(sqrt(22681873)*232562^(1/4)
*(6414867847*sqrt(2)*(x - 6) - 9072749200*x + 54436495200)*sqrt((6414867847*sqrt
(2) - 9072749200)/(58200487126974972400*sqrt(2) - 82307918517524735409)) + 44*sq
rt(22681873)*sqrt(2*x^2 - x + 3)*(85754*sqrt(2) - 123161))/(2*sqrt(22681873)*232
562^(1/4)*sqrt(31)*(6414867847*sqrt(2)*x - 9072749200*x)*sqrt(-sqrt(2)*(2*232562
^(1/4)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(198762031129681560785565546777639977*x + 82
329928978737020071375607715487825) - 281091960108418580856941154493127802*x - 11
6432102150944540714189939062152152)*sqrt((6414867847*sqrt(2) - 9072749200)/(5820
0487126974972400*sqrt(2) - 82307918517524735409)) + 2628482505413029835340603832
38400*x^2 + sqrt(2)*(73179342480249126097550902151600*x^2 - 10560328430724378104
64498574423*sqrt(2)*(49*x^2 - 151*x + 200) - 225511851316686082463881351528400*x
+ 298691193796935208561432253680000) - 92930890190374527320875874549224*sqrt(2)
*(2*x^2 - x + 3) - 131424125270651491767030191619200*x + 39427237581195447530109
0574857600)/(1056032843072437810464498574423*sqrt(2)*x^2 - 149345596898467604280
7161268400*x^2))*sqrt((6414867847*sqrt(2) - 9072749200)/(58200487126974972400*sq
rt(2) - 82307918517524735409)) + sqrt(22681873)*232562^(1/4)*sqrt(31)*(641486784
7*sqrt(2)*(19*x - 22) - 172382234800*x + 199600482400)*sqrt((6414867847*sqrt(2)
- 9072749200)/(58200487126974972400*sqrt(2) - 82307918517524735409)) - 1364*sqrt
(22681873)*sqrt(31)*sqrt(2*x^2 - x + 3)*(18658*sqrt(2) - 26103))) + 46113488900*
sqrt(22681873)*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(-31*(sqrt(22
681873)*232562^(1/4)*(6414867847*sqrt(2)*(x - 6) - 9072749200*x + 54436495200)*s
qrt((6414867847*sqrt(2) - 9072749200)/(58200487126974972400*sqrt(2) - 8230791851
7524735409)) - 44*sqrt(22681873)*sqrt(2*x^2 - x + 3)*(85754*sqrt(2) - 123161))/(
2*sqrt(22681873)*232562^(1/4)*sqrt(31)*(6414867847*sqrt(2)*x - 9072749200*x)*sqr
t(sqrt(2)*(2*232562^(1/4)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(198762031129681560785565
546777639977*x + 82329928978737020071375607715487825) - 281091960108418580856941
154493127802*x - 116432102150944540714189939062152152)*sqrt((6414867847*sqrt(2)
- 9072749200)/(58200487126974972400*sqrt(2) - 82307918517524735409)) - 262848250
541302983534060383238400*x^2 - sqrt(2)*(73179342480249126097550902151600*x^2 - 1
056032843072437810464498574423*sqrt(2)*(49*x^2 - 151*x + 200) - 2255118513166860
82463881351528400*x + 298691193796935208561432253680000) + 929308901903745273208
75874549224*sqrt(2)*(2*x^2 - x + 3) + 131424125270651491767030191619200*x - 3942
72375811954475301090574857600)/(1056032843072437810464498574423*sqrt(2)*x^2 - 14
93455968984676042807161268400*x^2))*sqrt((6414867847*sqrt(2) - 9072749200)/(5820
0487126974972400*sqrt(2) - 82307918517524735409)) + sqrt(22681873)*232562^(1/4)*
sqrt(31)*(6414867847*sqrt(2)*(19*x - 22) - 172382234800*x + 199600482400)*sqrt((
6414867847*sqrt(2) - 9072749200)/(58200487126974972400*sqrt(2) - 823079185175247
35409)) + 1364*sqrt(22681873)*sqrt(31)*sqrt(2*x^2 - x + 3)*(18658*sqrt(2) - 2610
3))) + 25*sqrt(22681873)*sqrt(31)*(226818730000*x^4 + 272182476000*x^3 + 2631097
26800*x^2 - 6414867847*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 108872990
400*x + 36290996800)*log(-803855254356451562500*sqrt(2)*(2*232562^(1/4)*sqrt(2*x
^2 - x + 3)*(sqrt(2)*(198762031129681560785565546777639977*x + 82329928978737020
071375607715487825) - 281091960108418580856941154493127802*x - 11643210215094454
0714189939062152152)*sqrt((6414867847*sqrt(2) - 9072749200)/(5820048712697497240
0*sqrt(2) - 82307918517524735409)) + 262848250541302983534060383238400*x^2 + sqr
t(2)*(73179342480249126097550902151600*x^2 - 1056032843072437810464498574423*sqr
t(2)*(49*x^2 - 151*x + 200) - 225511851316686082463881351528400*x + 298691193796
935208561432253680000) - 92930890190374527320875874549224*sqrt(2)*(2*x^2 - x + 3
) - 131424125270651491767030191619200*x + 394272375811954475301090574857600)/(10
56032843072437810464498574423*sqrt(2)*x^2 - 1493455968984676042807161268400*x^2)
) - 25*sqrt(22681873)*sqrt(31)*(226818730000*x^4 + 272182476000*x^3 + 2631097268
00*x^2 - 6414867847*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 108872990400
*x + 36290996800)*log(803855254356451562500*sqrt(2)*(2*232562^(1/4)*sqrt(2*x^2 -
x + 3)*(sqrt(2)*(198762031129681560785565546777639977*x + 823299289787370200713
75607715487825) - 281091960108418580856941154493127802*x - 116432102150944540714
189939062152152)*sqrt((6414867847*sqrt(2) - 9072749200)/(58200487126974972400*sq
rt(2) - 82307918517524735409)) - 262848250541302983534060383238400*x^2 - sqrt(2)
*(73179342480249126097550902151600*x^2 - 1056032843072437810464498574423*sqrt(2)
*(49*x^2 - 151*x + 200) - 225511851316686082463881351528400*x + 2986911937969352
08561432253680000) + 92930890190374527320875874549224*sqrt(2)*(2*x^2 - x + 3) +
131424125270651491767030191619200*x - 394272375811954475301090574857600)/(105603
2843072437810464498574423*sqrt(2)*x^2 - 1493455968984676042807161268400*x^2)))/(
(226818730000*x^4 + 272182476000*x^3 + 263109726800*x^2 - 6414867847*sqrt(2)*(25
*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 108872990400*x + 36290996800)*sqrt((6414867
847*sqrt(2) - 9072749200)/(58200487126974972400*sqrt(2) - 82307918517524735409))
)
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(5*x**2+3*x+2)**3/(2*x**2-x+3)**(1/2),x)
[Out]
Integral(1/(sqrt(2*x**2 - x + 3)*(5*x**2 + 3*x + 2)**3), x)
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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(1/((5*x^2 + 3*x + 2)^3*sqrt(2*x^2 - x + 3)),x, algorithm="giac")
[Out]
Exception raised: RuntimeError