3.71 \(\int \frac{\left (3-x+2 x^2\right )^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx\)
Optimal. Leaf size=223 \[ \frac{(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac{3 (696 x+277) \sqrt{2 x^2-x+3}}{3844 \left (5 x^2+3 x+2\right )}+\frac{3 \sqrt{\frac{1}{682} \left (366990269+259509026 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (366990269+259509026 \sqrt{2}\right )}} \left (\left (70517+49942 \sqrt{2}\right ) x+20575 \sqrt{2}+29367\right )}{\sqrt{2 x^2-x+3}}\right )}{7688}-\frac{3 \sqrt{\frac{1}{682} \left (259509026 \sqrt{2}-366990269\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (259509026 \sqrt{2}-366990269\right )}} \left (\left (70517-49942 \sqrt{2}\right ) x-20575 \sqrt{2}+29367\right )}{\sqrt{2 x^2-x+3}}\right )}{7688} \]
[Out]
((3 + 10*x)*(3 - x + 2*x^2)^(3/2))/(62*(2 + 3*x + 5*x^2)^2) + (3*(277 + 696*x)*S
qrt[3 - x + 2*x^2])/(3844*(2 + 3*x + 5*x^2)) + (3*Sqrt[(366990269 + 259509026*Sq
rt[2])/682]*ArcTan[(Sqrt[11/(31*(366990269 + 259509026*Sqrt[2]))]*(29367 + 20575
*Sqrt[2] + (70517 + 49942*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/7688 - (3*Sqrt[(-36
6990269 + 259509026*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-366990269 + 259509026*S
qrt[2]))]*(29367 - 20575*Sqrt[2] + (70517 - 49942*Sqrt[2])*x))/Sqrt[3 - x + 2*x^
2]])/7688
_______________________________________________________________________________________
Rubi [A] time = 0.911228, 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
\[ \frac{(10 x+3) \left (2 x^2-x+3\right )^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac{3 (696 x+277) \sqrt{2 x^2-x+3}}{3844 \left (5 x^2+3 x+2\right )}+\frac{3 \sqrt{\frac{1}{682} \left (366990269+259509026 \sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (366990269+259509026 \sqrt{2}\right )}} \left (\left (70517+49942 \sqrt{2}\right ) x+20575 \sqrt{2}+29367\right )}{\sqrt{2 x^2-x+3}}\right )}{7688}-\frac{3 \sqrt{\frac{1}{682} \left (259509026 \sqrt{2}-366990269\right )} \tanh ^{-1}\left (\frac{\sqrt{\frac{11}{31 \left (259509026 \sqrt{2}-366990269\right )}} \left (\left (70517-49942 \sqrt{2}\right ) x-20575 \sqrt{2}+29367\right )}{\sqrt{2 x^2-x+3}}\right )}{7688} \]
Antiderivative was successfully verified.
[In] Int[(3 - x + 2*x^2)^(3/2)/(2 + 3*x + 5*x^2)^3,x]
[Out]
((3 + 10*x)*(3 - x + 2*x^2)^(3/2))/(62*(2 + 3*x + 5*x^2)^2) + (3*(277 + 696*x)*S
qrt[3 - x + 2*x^2])/(3844*(2 + 3*x + 5*x^2)) + (3*Sqrt[(366990269 + 259509026*Sq
rt[2])/682]*ArcTan[(Sqrt[11/(31*(366990269 + 259509026*Sqrt[2]))]*(29367 + 20575
*Sqrt[2] + (70517 + 49942*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]])/7688 - (3*Sqrt[(-36
6990269 + 259509026*Sqrt[2])/682]*ArcTanh[(Sqrt[11/(31*(-366990269 + 259509026*S
qrt[2]))]*(29367 - 20575*Sqrt[2] + (70517 - 49942*Sqrt[2])*x))/Sqrt[3 - x + 2*x^
2]])/7688
_______________________________________________________________________________________
Rubi in Sympy [A] time = 92.9457, size = 284, normalized size = 1.27 \[ \frac{\left (10 x + 3\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}{62 \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{\left (1044 x + \frac{831}{2}\right ) \sqrt{2 x^{2} - x + 3}}{1922 \left (5 x^{2} + 3 x + 2\right )} + \frac{1404928 \cdot 2^{\frac{47}{114}} \cdot 3^{\frac{3}{19}} \sqrt{341} \cdot 5^{\frac{32}{57}} \cdot 7^{\frac{40}{57}} \left (\frac{206481}{2} + \frac{146949 \sqrt{2}}{2}\right ) \left (\frac{678975 \sqrt{2}}{4} + \frac{969111}{4}\right ) \operatorname{atan}{\left (- \frac{5619712 \cdot 2^{\frac{52}{57}} \cdot 3^{\frac{3}{19}} \sqrt{341} \cdot 5^{\frac{32}{57}} \cdot 7^{\frac{40}{57}} \left (x \left (- \frac{824043 \sqrt{2}}{2} - \frac{2327061}{4}\right ) - \frac{969111}{4} - \frac{678975 \sqrt{2}}{4}\right )}{842926025390625 \sqrt{2 x^{2} - x + 3}} \right )}}{98016281158447265625} - \frac{\sqrt{682} \left (- \frac{678975 \sqrt{2}}{4} + \frac{969111}{4}\right ) \left (- \frac{146949 \sqrt{2}}{2} + \frac{206481}{2}\right ) \operatorname{atanh}{\left (\frac{4 \sqrt{341} \left (x \left (- \frac{2327061}{4} + \frac{824043 \sqrt{2}}{2}\right ) - \frac{969111}{4} + \frac{678975 \sqrt{2}}{4}\right )}{1023 \sqrt{-366990269 + 259509026 \sqrt{2}} \sqrt{2 x^{2} - x + 3}} \right )}}{237910926 \sqrt{-366990269 + 259509026 \sqrt{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*x**2-x+3)**(3/2)/(5*x**2+3*x+2)**3,x)
[Out]
(10*x + 3)*(2*x**2 - x + 3)**(3/2)/(62*(5*x**2 + 3*x + 2)**2) + (1044*x + 831/2)
*sqrt(2*x**2 - x + 3)/(1922*(5*x**2 + 3*x + 2)) + 1404928*2**(47/114)*3**(3/19)*
sqrt(341)*5**(32/57)*7**(40/57)*(206481/2 + 146949*sqrt(2)/2)*(678975*sqrt(2)/4
+ 969111/4)*atan(-5619712*2**(52/57)*3**(3/19)*sqrt(341)*5**(32/57)*7**(40/57)*(
x*(-824043*sqrt(2)/2 - 2327061/4) - 969111/4 - 678975*sqrt(2)/4)/(84292602539062
5*sqrt(2*x**2 - x + 3)))/98016281158447265625 - sqrt(682)*(-678975*sqrt(2)/4 + 9
69111/4)*(-146949*sqrt(2)/2 + 206481/2)*atanh(4*sqrt(341)*(x*(-2327061/4 + 82404
3*sqrt(2)/2) - 969111/4 + 678975*sqrt(2)/4)/(1023*sqrt(-366990269 + 259509026*sq
rt(2))*sqrt(2*x**2 - x + 3)))/(237910926*sqrt(-366990269 + 259509026*sqrt(2)))
_______________________________________________________________________________________
Mathematica [C] time = 6.57501, size = 1171, normalized size = 5.25 \[ \sqrt{2 x^2-x+3} \left (\frac{11 (13 x+7)}{310 \left (5 x^2+3 x+2\right )^2}+\frac{11680 x+3163}{19220 \left (5 x^2+3 x+2\right )}\right )-\frac{3 i \left (-24971 i+902 \sqrt{31}\right ) \tan ^{-1}\left (\frac{31 \left (1789928800 \sqrt{31} x^4+56810945600 i x^4+10089483360 \sqrt{31} x^3-151681537680 i x^3+14045028558 \sqrt{31} x^2+158238605196 i x^2+8050492021 \sqrt{31} x-148151300773 i x+25278538857 \sqrt{31}+31227856109 i\right )}{-1305722486200 i \sqrt{31} x^4+1536126024400 x^4+64877256500 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3-535663546990 i \sqrt{31} x^3+4234217180380 x^3+22707039775 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2-1457613959802 i \sqrt{31} x^2+689282588324 x^2+16219314125 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x-560818641999 i \sqrt{31} x+7060303464863 x-6487725650 i \sqrt{682 \left (13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}+251835138467 i \sqrt{31}+1329350472021}\right )}{3844 \sqrt{682 \left (13+i \sqrt{31}\right )}}-\frac{3 i \left (24971 i+902 \sqrt{31}\right ) \tanh ^{-1}\left (\frac{11 \left (5044344800 \sqrt{31} x^4+562393146150 i x^4+28433998560 \sqrt{31} x^3-1365505300720 i x^3+39581444118 \sqrt{31} x^2+1456138041834 i x^2+22687750241 \sqrt{31} x+1296309231133 i x+71239518597 \sqrt{31}+1091580705511 i\right )}{1305722486200 \sqrt{31} x^4-1536126024400 i x^4+356824910750 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^3+535663546990 \sqrt{31} x^3-4234217180380 i x^3-807721843425 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x^2+1457613959802 \sqrt{31} x^2-689282588324 i x^2-470360109625 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3} x+560818641999 \sqrt{31} x-7060303464863 i x-408726715950 \sqrt{22 \left (-13+i \sqrt{31}\right )} \sqrt{2 x^2-x+3}-251835138467 \sqrt{31}-1329350472021 i}\right )}{3844 \sqrt{682 \left (-13+i \sqrt{31}\right )}}+\frac{3 i \left (24971 i+902 \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 )}{7688 \sqrt{682 \left (-13+i \sqrt{31}\right )}}-\frac{3 \left (-24971 i+902 \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 )}{7688 \sqrt{682 \left (13+i \sqrt{31}\right )}}-\frac{3 i \left (24971 i+902 \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 )}{7688 \sqrt{682 \left (-13+i \sqrt{31}\right )}}+\frac{3 \left (-24971 i+902 \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 )}{7688 \sqrt{682 \left (13+i \sqrt{31}\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(3 - x + 2*x^2)^(3/2)/(2 + 3*x + 5*x^2)^3,x]
[Out]
Sqrt[3 - x + 2*x^2]*((11*(7 + 13*x))/(310*(2 + 3*x + 5*x^2)^2) + (3163 + 11680*x
)/(19220*(2 + 3*x + 5*x^2))) - (((3*I)/3844)*(-24971*I + 902*Sqrt[31])*ArcTan[(3
1*(31227856109*I + 25278538857*Sqrt[31] - (148151300773*I)*x + 8050492021*Sqrt[3
1]*x + (158238605196*I)*x^2 + 14045028558*Sqrt[31]*x^2 - (151681537680*I)*x^3 +
10089483360*Sqrt[31]*x^3 + (56810945600*I)*x^4 + 1789928800*Sqrt[31]*x^4))/(1329
350472021 + (251835138467*I)*Sqrt[31] + 7060303464863*x - (560818641999*I)*Sqrt[
31]*x + 689282588324*x^2 - (1457613959802*I)*Sqrt[31]*x^2 + 4234217180380*x^3 -
(535663546990*I)*Sqrt[31]*x^3 + 1536126024400*x^4 - (1305722486200*I)*Sqrt[31]*x
^4 - (6487725650*I)*Sqrt[682*(13 + I*Sqrt[31])]*Sqrt[3 - x + 2*x^2] + (162193141
25*I)*Sqrt[682*(13 + I*Sqrt[31])]*x*Sqrt[3 - x + 2*x^2] + (22707039775*I)*Sqrt[6
82*(13 + I*Sqrt[31])]*x^2*Sqrt[3 - x + 2*x^2] + (64877256500*I)*Sqrt[682*(13 + I
*Sqrt[31])]*x^3*Sqrt[3 - x + 2*x^2])])/Sqrt[682*(13 + I*Sqrt[31])] - (((3*I)/384
4)*(24971*I + 902*Sqrt[31])*ArcTanh[(11*(1091580705511*I + 71239518597*Sqrt[31]
+ (1296309231133*I)*x + 22687750241*Sqrt[31]*x + (1456138041834*I)*x^2 + 3958144
4118*Sqrt[31]*x^2 - (1365505300720*I)*x^3 + 28433998560*Sqrt[31]*x^3 + (56239314
6150*I)*x^4 + 5044344800*Sqrt[31]*x^4))/(-1329350472021*I - 251835138467*Sqrt[31
] - (7060303464863*I)*x + 560818641999*Sqrt[31]*x - (689282588324*I)*x^2 + 14576
13959802*Sqrt[31]*x^2 - (4234217180380*I)*x^3 + 535663546990*Sqrt[31]*x^3 - (153
6126024400*I)*x^4 + 1305722486200*Sqrt[31]*x^4 - 408726715950*Sqrt[22*(-13 + I*S
qrt[31])]*Sqrt[3 - x + 2*x^2] - 470360109625*Sqrt[22*(-13 + I*Sqrt[31])]*x*Sqrt[
3 - x + 2*x^2] - 807721843425*Sqrt[22*(-13 + I*Sqrt[31])]*x^2*Sqrt[3 - x + 2*x^2
] + 356824910750*Sqrt[22*(-13 + I*Sqrt[31])]*x^3*Sqrt[3 - x + 2*x^2])])/Sqrt[682
*(-13 + I*Sqrt[31])] - (3*(-24971*I + 902*Sqrt[31])*Log[(-3*I + Sqrt[31] - (10*I
)*x)^2*(3*I + Sqrt[31] + (10*I)*x)^2])/(7688*Sqrt[682*(13 + I*Sqrt[31])]) + (((3
*I)/7688)*(24971*I + 902*Sqrt[31])*Log[(-3*I + Sqrt[31] - (10*I)*x)^2*(3*I + Sqr
t[31] + (10*I)*x)^2])/Sqrt[682*(-13 + I*Sqrt[31])] - (((3*I)/7688)*(24971*I + 90
2*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])] + (3*(-24971*I + 902*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*Sq
rt[31])]*x*Sqrt[3 - x + 2*x^2])])/(7688*Sqrt[682*(13 + I*Sqrt[31])])
_______________________________________________________________________________________
Maple [B] time = 0.373, size = 81415, normalized size = 365.1 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*x^2-x+3)^(3/2)/(5*x^2+3*x+2)^3,x)
[Out]
result too large to display
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 - x + 3)^(3/2)/(5*x^2 + 3*x + 2)^3,x, algorithm="maxima")
[Out]
integrate((2*x^2 - x + 3)^(3/2)/(5*x^2 + 3*x + 2)^3, x)
_______________________________________________________________________________________
Fricas [A] time = 0.367025, size = 1601, normalized size = 7.18 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x^2 - x + 3)^(3/2)/(5*x^2 + 3*x + 2)^3,x, algorithm="fricas")
[Out]
1/28767113409191937472*sqrt(129754513)*232562^(3/4)*sqrt(31)*(8*sqrt(129754513)*
232562^(1/4)*sqrt(31)*(6062130847360*x^3 + 5278932606892*x^2 - 366990269*sqrt(2)
*(11680*x^3 + 10171*x^2 + 8343*x + 2220) + 4330167607836*x + 1152220075440)*sqrt
(2*x^2 - x + 3)*sqrt((366990269*sqrt(2) - 519018052)/(190474574519335988*sqrt(2)
- 269371726691629713)) + 189113268*sqrt(129754513)*sqrt(2)*(25*x^4 + 30*x^3 + 2
9*x^2 + 12*x + 4)*arctan(31*(sqrt(129754513)*232562^(1/4)*(366990269*sqrt(2)*(x
- 6) - 519018052*x + 3114108312)*sqrt((366990269*sqrt(2) - 519018052)/(190474574
519335988*sqrt(2) - 269371726691629713)) + 44*sqrt(129754513)*sqrt(2*x^2 - x + 3
)*(20575*sqrt(2) - 29367))/(2*sqrt(129754513)*232562^(1/4)*sqrt(31)*(366990269*s
qrt(2)*x - 519018052*x)*sqrt(-sqrt(2)*(46*232562^(1/4)*sqrt(2*x^2 - x + 3)*(sqrt
(2)*(386989138339976299220055293849*x + 160296149591496060582087946227) - 547285
287931472359802143240076*x - 226692988748480238637967347622)*sqrt((366990269*sqr
t(2) - 519018052)/(190474574519335988*sqrt(2) - 269371726691629713)) + 492119618
37700530566998189120*x^2 + sqrt(2)*(13701057557087079532857450380*x^2 - 19771654
5062110269247518173*sqrt(2)*(49*x^2 - 151*x + 200) - 422216263493907961114586736
20*x + 55922683906477875644316124000) - 17399055965465703693781599224*sqrt(2)*(2
*x^2 - x + 3) - 24605980918850265283499094560*x + 73817942756550795850497283680)
/(197716545062110269247518173*sqrt(2)*x^2 - 279613419532389378221580620*x^2))*sq
rt((366990269*sqrt(2) - 519018052)/(190474574519335988*sqrt(2) - 269371726691629
713)) + sqrt(129754513)*232562^(1/4)*sqrt(31)*(366990269*sqrt(2)*(19*x - 22) - 9
861342988*x + 11418397144)*sqrt((366990269*sqrt(2) - 519018052)/(190474574519335
988*sqrt(2) - 269371726691629713)) - 1364*sqrt(129754513)*sqrt(31)*sqrt(2*x^2 -
x + 3)*(4453*sqrt(2) - 6257))) + 189113268*sqrt(129754513)*sqrt(2)*(25*x^4 + 30*
x^3 + 29*x^2 + 12*x + 4)*arctan(-31*(sqrt(129754513)*232562^(1/4)*(366990269*sqr
t(2)*(x - 6) - 519018052*x + 3114108312)*sqrt((366990269*sqrt(2) - 519018052)/(1
90474574519335988*sqrt(2) - 269371726691629713)) - 44*sqrt(129754513)*sqrt(2*x^2
- x + 3)*(20575*sqrt(2) - 29367))/(2*sqrt(129754513)*232562^(1/4)*sqrt(31)*(366
990269*sqrt(2)*x - 519018052*x)*sqrt(sqrt(2)*(46*232562^(1/4)*sqrt(2*x^2 - x + 3
)*(sqrt(2)*(386989138339976299220055293849*x + 160296149591496060582087946227) -
547285287931472359802143240076*x - 226692988748480238637967347622)*sqrt((366990
269*sqrt(2) - 519018052)/(190474574519335988*sqrt(2) - 269371726691629713)) - 49
211961837700530566998189120*x^2 - sqrt(2)*(13701057557087079532857450380*x^2 - 1
97716545062110269247518173*sqrt(2)*(49*x^2 - 151*x + 200) - 42221626349390796111
458673620*x + 55922683906477875644316124000) + 17399055965465703693781599224*sqr
t(2)*(2*x^2 - x + 3) + 24605980918850265283499094560*x - 73817942756550795850497
283680)/(197716545062110269247518173*sqrt(2)*x^2 - 279613419532389378221580620*x
^2))*sqrt((366990269*sqrt(2) - 519018052)/(190474574519335988*sqrt(2) - 26937172
6691629713)) + sqrt(129754513)*232562^(1/4)*sqrt(31)*(366990269*sqrt(2)*(19*x -
22) - 9861342988*x + 11418397144)*sqrt((366990269*sqrt(2) - 519018052)/(19047457
4519335988*sqrt(2) - 269371726691629713)) + 1364*sqrt(129754513)*sqrt(31)*sqrt(2
*x^2 - x + 3)*(4453*sqrt(2) - 6257))) + 3*sqrt(129754513)*sqrt(31)*(12975451300*
x^4 + 15570541560*x^3 + 15051523508*x^2 - 366990269*sqrt(2)*(25*x^4 + 30*x^3 + 2
9*x^2 + 12*x + 4) + 6228216624*x + 2076072208)*log(-606104411179218084*sqrt(2)*(
46*232562^(1/4)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(386989138339976299220055293849*x +
160296149591496060582087946227) - 547285287931472359802143240076*x - 2266929887
48480238637967347622)*sqrt((366990269*sqrt(2) - 519018052)/(190474574519335988*s
qrt(2) - 269371726691629713)) + 49211961837700530566998189120*x^2 + sqrt(2)*(137
01057557087079532857450380*x^2 - 197716545062110269247518173*sqrt(2)*(49*x^2 - 1
51*x + 200) - 42221626349390796111458673620*x + 55922683906477875644316124000) -
17399055965465703693781599224*sqrt(2)*(2*x^2 - x + 3) - 24605980918850265283499
094560*x + 73817942756550795850497283680)/(197716545062110269247518173*sqrt(2)*x
^2 - 279613419532389378221580620*x^2)) - 3*sqrt(129754513)*sqrt(31)*(12975451300
*x^4 + 15570541560*x^3 + 15051523508*x^2 - 366990269*sqrt(2)*(25*x^4 + 30*x^3 +
29*x^2 + 12*x + 4) + 6228216624*x + 2076072208)*log(606104411179218084*sqrt(2)*(
46*232562^(1/4)*sqrt(2*x^2 - x + 3)*(sqrt(2)*(386989138339976299220055293849*x +
160296149591496060582087946227) - 547285287931472359802143240076*x - 2266929887
48480238637967347622)*sqrt((366990269*sqrt(2) - 519018052)/(190474574519335988*s
qrt(2) - 269371726691629713)) - 49211961837700530566998189120*x^2 - sqrt(2)*(137
01057557087079532857450380*x^2 - 197716545062110269247518173*sqrt(2)*(49*x^2 - 1
51*x + 200) - 42221626349390796111458673620*x + 55922683906477875644316124000) +
17399055965465703693781599224*sqrt(2)*(2*x^2 - x + 3) + 24605980918850265283499
094560*x - 73817942756550795850497283680)/(197716545062110269247518173*sqrt(2)*x
^2 - 279613419532389378221580620*x^2)))/((12975451300*x^4 + 15570541560*x^3 + 15
051523508*x^2 - 366990269*sqrt(2)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 622821
6624*x + 2076072208)*sqrt((366990269*sqrt(2) - 519018052)/(190474574519335988*sq
rt(2) - 269371726691629713)))
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}{\left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x**2-x+3)**(3/2)/(5*x**2+3*x+2)**3,x)
[Out]
Integral((2*x**2 - x + 3)**(3/2)/(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((2*x^2 - x + 3)^(3/2)/(5*x^2 + 3*x + 2)^3,x, algorithm="giac")
[Out]
Exception raised: RuntimeError