3.47 \(\int \frac{1}{(3-2 x)^{11/2} \left (1+x+2 x^2\right )^5} \, dx\)

Optimal. Leaf size=407 \[ \frac{x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}+\frac{5 (4377 x+3049)}{153664 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}+\frac{3049 x+1387}{32928 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}+\frac{73 x+23}{1176 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}-\frac{38225}{240945152 \sqrt{3-2 x}}-\frac{141045}{120472576 (3-2 x)^{3/2}}-\frac{38491}{8605184 (3-2 x)^{5/2}}-\frac{462025}{30118144 (3-2 x)^{7/2}}-\frac{19255}{395136 (3-2 x)^{9/2}}+\frac{5 \sqrt{\frac{1}{2} \left (40815066112 \sqrt{14}-149046503977\right )} \log \left (-2 x-\sqrt{7+2 \sqrt{14}} \sqrt{3-2 x}+\sqrt{14}+3\right )}{6746464256}-\frac{5 \sqrt{\frac{1}{2} \left (40815066112 \sqrt{14}-149046503977\right )} \log \left (-2 x+\sqrt{7+2 \sqrt{14}} \sqrt{3-2 x}+\sqrt{14}+3\right )}{6746464256}+\frac{5 \sqrt{\frac{1}{2} \left (149046503977+40815066112 \sqrt{14}\right )} \tan ^{-1}\left (\frac{\sqrt{7+2 \sqrt{14}}-2 \sqrt{3-2 x}}{\sqrt{2 \sqrt{14}-7}}\right )}{3373232128}-\frac{5 \sqrt{\frac{1}{2} \left (149046503977+40815066112 \sqrt{14}\right )} \tan ^{-1}\left (\frac{2 \sqrt{3-2 x}+\sqrt{7+2 \sqrt{14}}}{\sqrt{2 \sqrt{14}-7}}\right )}{3373232128} \]

[Out]

-19255/(395136*(3 - 2*x)^(9/2)) - 462025/(30118144*(3 - 2*x)^(7/2)) - 38491/(860
5184*(3 - 2*x)^(5/2)) - 141045/(120472576*(3 - 2*x)^(3/2)) - 38225/(240945152*Sq
rt[3 - 2*x]) + x/(28*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)^4) + (23 + 73*x)/(1176*(3 -
2*x)^(9/2)*(1 + x + 2*x^2)^3) + (1387 + 3049*x)/(32928*(3 - 2*x)^(9/2)*(1 + x +
2*x^2)^2) + (5*(3049 + 4377*x))/(153664*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)) + (5*S
qrt[(149046503977 + 40815066112*Sqrt[14])/2]*ArcTan[(Sqrt[7 + 2*Sqrt[14]] - 2*Sq
rt[3 - 2*x])/Sqrt[-7 + 2*Sqrt[14]]])/3373232128 - (5*Sqrt[(149046503977 + 408150
66112*Sqrt[14])/2]*ArcTan[(Sqrt[7 + 2*Sqrt[14]] + 2*Sqrt[3 - 2*x])/Sqrt[-7 + 2*S
qrt[14]]])/3373232128 + (5*Sqrt[(-149046503977 + 40815066112*Sqrt[14])/2]*Log[3
+ Sqrt[14] - Sqrt[7 + 2*Sqrt[14]]*Sqrt[3 - 2*x] - 2*x])/6746464256 - (5*Sqrt[(-1
49046503977 + 40815066112*Sqrt[14])/2]*Log[3 + Sqrt[14] + Sqrt[7 + 2*Sqrt[14]]*S
qrt[3 - 2*x] - 2*x])/6746464256

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Rubi [A]  time = 1.47675, antiderivative size = 407, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ \frac{x}{28 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}+\frac{5 (4377 x+3049)}{153664 (3-2 x)^{9/2} \left (2 x^2+x+1\right )}+\frac{3049 x+1387}{32928 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^2}+\frac{73 x+23}{1176 (3-2 x)^{9/2} \left (2 x^2+x+1\right )^3}-\frac{38225}{240945152 \sqrt{3-2 x}}-\frac{141045}{120472576 (3-2 x)^{3/2}}-\frac{38491}{8605184 (3-2 x)^{5/2}}-\frac{462025}{30118144 (3-2 x)^{7/2}}-\frac{19255}{395136 (3-2 x)^{9/2}}+\frac{5 \sqrt{\frac{1}{2} \left (40815066112 \sqrt{14}-149046503977\right )} \log \left (-2 x-\sqrt{7+2 \sqrt{14}} \sqrt{3-2 x}+\sqrt{14}+3\right )}{6746464256}-\frac{5 \sqrt{\frac{1}{2} \left (40815066112 \sqrt{14}-149046503977\right )} \log \left (-2 x+\sqrt{7+2 \sqrt{14}} \sqrt{3-2 x}+\sqrt{14}+3\right )}{6746464256}+\frac{5 \sqrt{\frac{1}{2} \left (149046503977+40815066112 \sqrt{14}\right )} \tan ^{-1}\left (\frac{\sqrt{7+2 \sqrt{14}}-2 \sqrt{3-2 x}}{\sqrt{2 \sqrt{14}-7}}\right )}{3373232128}-\frac{5 \sqrt{\frac{1}{2} \left (149046503977+40815066112 \sqrt{14}\right )} \tan ^{-1}\left (\frac{2 \sqrt{3-2 x}+\sqrt{7+2 \sqrt{14}}}{\sqrt{2 \sqrt{14}-7}}\right )}{3373232128} \]

Antiderivative was successfully verified.

[In]  Int[1/((3 - 2*x)^(11/2)*(1 + x + 2*x^2)^5),x]

[Out]

-19255/(395136*(3 - 2*x)^(9/2)) - 462025/(30118144*(3 - 2*x)^(7/2)) - 38491/(860
5184*(3 - 2*x)^(5/2)) - 141045/(120472576*(3 - 2*x)^(3/2)) - 38225/(240945152*Sq
rt[3 - 2*x]) + x/(28*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)^4) + (23 + 73*x)/(1176*(3 -
2*x)^(9/2)*(1 + x + 2*x^2)^3) + (1387 + 3049*x)/(32928*(3 - 2*x)^(9/2)*(1 + x +
2*x^2)^2) + (5*(3049 + 4377*x))/(153664*(3 - 2*x)^(9/2)*(1 + x + 2*x^2)) + (5*S
qrt[(149046503977 + 40815066112*Sqrt[14])/2]*ArcTan[(Sqrt[7 + 2*Sqrt[14]] - 2*Sq
rt[3 - 2*x])/Sqrt[-7 + 2*Sqrt[14]]])/3373232128 - (5*Sqrt[(149046503977 + 408150
66112*Sqrt[14])/2]*ArcTan[(Sqrt[7 + 2*Sqrt[14]] + 2*Sqrt[3 - 2*x])/Sqrt[-7 + 2*S
qrt[14]]])/3373232128 + (5*Sqrt[(-149046503977 + 40815066112*Sqrt[14])/2]*Log[3
+ Sqrt[14] - Sqrt[7 + 2*Sqrt[14]]*Sqrt[3 - 2*x] - 2*x])/6746464256 - (5*Sqrt[(-1
49046503977 + 40815066112*Sqrt[14])/2]*Log[3 + Sqrt[14] + Sqrt[7 + 2*Sqrt[14]]*S
qrt[3 - 2*x] - 2*x])/6746464256

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 48.1711, size = 478, normalized size = 1.17 \[ \frac{x}{28 \left (- 2 x + 3\right )^{\frac{9}{2}} \left (2 x^{2} + x + 1\right )^{4}} + \frac{\sqrt{14} \left (- 48353174284800 \sqrt{14} + 732027212367360\right ) \log{\left (- 2 x - \sqrt{7 + 2 \sqrt{14}} \sqrt{- 2 x + 3} + 3 + \sqrt{14} \right )}}{17068042484057112576 \sqrt{7 + 2 \sqrt{14}}} - \frac{\sqrt{14} \left (- 48353174284800 \sqrt{14} + 732027212367360\right ) \log{\left (- 2 x + \sqrt{7 + 2 \sqrt{14}} \sqrt{- 2 x + 3} + 3 + \sqrt{14} \right )}}{17068042484057112576 \sqrt{7 + 2 \sqrt{14}}} + \frac{\sqrt{14} \left (- 1464054424734720 \sqrt{7 + 2 \sqrt{14}} + \frac{\sqrt{7 + 2 \sqrt{14}} \left (- 96706348569600 \sqrt{14} + 1464054424734720\right )}{2}\right ) \operatorname{atan}{\left (\frac{2 \sqrt{- 2 x + 3} - \sqrt{7 + 2 \sqrt{14}}}{\sqrt{-7 + 2 \sqrt{14}}} \right )}}{8534021242028556288 \sqrt{-7 + 2 \sqrt{14}} \sqrt{7 + 2 \sqrt{14}}} + \frac{\sqrt{14} \left (- 1464054424734720 \sqrt{7 + 2 \sqrt{14}} + \frac{\sqrt{7 + 2 \sqrt{14}} \left (- 96706348569600 \sqrt{14} + 1464054424734720\right )}{2}\right ) \operatorname{atan}{\left (\frac{2 \sqrt{- 2 x + 3} + \sqrt{7 + 2 \sqrt{14}}}{\sqrt{-7 + 2 \sqrt{14}}} \right )}}{8534021242028556288 \sqrt{-7 + 2 \sqrt{14}} \sqrt{7 + 2 \sqrt{14}}} - \frac{38225}{240945152 \sqrt{- 2 x + 3}} - \frac{141045}{120472576 \left (- 2 x + 3\right )^{\frac{3}{2}}} - \frac{38491}{8605184 \left (- 2 x + 3\right )^{\frac{5}{2}}} - \frac{462025}{30118144 \left (- 2 x + 3\right )^{\frac{7}{2}}} + \frac{28616 x + 9016}{460992 \left (- 2 x + 3\right )^{\frac{9}{2}} \left (2 x^{2} + x + 1\right )^{3}} + \frac{16732912 x + 7611856}{180708864 \left (- 2 x + 3\right )^{\frac{9}{2}} \left (2 x^{2} + x + 1\right )^{2}} + \frac{5044404960 x + 3513911520}{35418937344 \left (- 2 x + 3\right )^{\frac{9}{2}} \left (2 x^{2} + x + 1\right )} - \frac{19255}{395136 \left (- 2 x + 3\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3-2*x)**(11/2)/(2*x**2+x+1)**5,x)

[Out]

x/(28*(-2*x + 3)**(9/2)*(2*x**2 + x + 1)**4) + sqrt(14)*(-48353174284800*sqrt(14
) + 732027212367360)*log(-2*x - sqrt(7 + 2*sqrt(14))*sqrt(-2*x + 3) + 3 + sqrt(1
4))/(17068042484057112576*sqrt(7 + 2*sqrt(14))) - sqrt(14)*(-48353174284800*sqrt
(14) + 732027212367360)*log(-2*x + sqrt(7 + 2*sqrt(14))*sqrt(-2*x + 3) + 3 + sqr
t(14))/(17068042484057112576*sqrt(7 + 2*sqrt(14))) + sqrt(14)*(-1464054424734720
*sqrt(7 + 2*sqrt(14)) + sqrt(7 + 2*sqrt(14))*(-96706348569600*sqrt(14) + 1464054
424734720)/2)*atan((2*sqrt(-2*x + 3) - sqrt(7 + 2*sqrt(14)))/sqrt(-7 + 2*sqrt(14
)))/(8534021242028556288*sqrt(-7 + 2*sqrt(14))*sqrt(7 + 2*sqrt(14))) + sqrt(14)*
(-1464054424734720*sqrt(7 + 2*sqrt(14)) + sqrt(7 + 2*sqrt(14))*(-96706348569600*
sqrt(14) + 1464054424734720)/2)*atan((2*sqrt(-2*x + 3) + sqrt(7 + 2*sqrt(14)))/s
qrt(-7 + 2*sqrt(14)))/(8534021242028556288*sqrt(-7 + 2*sqrt(14))*sqrt(7 + 2*sqrt
(14))) - 38225/(240945152*sqrt(-2*x + 3)) - 141045/(120472576*(-2*x + 3)**(3/2))
- 38491/(8605184*(-2*x + 3)**(5/2)) - 462025/(30118144*(-2*x + 3)**(7/2)) + (28
616*x + 9016)/(460992*(-2*x + 3)**(9/2)*(2*x**2 + x + 1)**3) + (16732912*x + 761
1856)/(180708864*(-2*x + 3)**(9/2)*(2*x**2 + x + 1)**2) + (5044404960*x + 351391
1520)/(35418937344*(-2*x + 3)**(9/2)*(2*x**2 + x + 1)) - 19255/(395136*(-2*x + 3
)**(9/2))

_______________________________________________________________________________________

Mathematica [C]  time = 2.25894, size = 206, normalized size = 0.51 \[ \frac{\frac{45 i \left (284993 \sqrt{7}+53515 i\right ) \tan ^{-1}\left (\frac{\sqrt{6-4 x}}{\sqrt{-7-i \sqrt{7}}}\right )}{\sqrt{-\frac{1}{2} i \left (\sqrt{7}-7 i\right )}}-\frac{45 i \left (284993 \sqrt{7}-53515 i\right ) \tan ^{-1}\left (\frac{\sqrt{6-4 x}}{\sqrt{-7+i \sqrt{7}}}\right )}{\sqrt{\frac{1}{2} i \left (\sqrt{7}+7 i\right )}}-\frac{14 \left (88070400 x^{12}-677249280 x^{11}[phone] x^{10}-2443779648 x^9+2343370048 x^8-3106712560 x^7+2888625656 x^6-1470758860 x^5+1627773523 x^4-1073855156 x^3+135202154 x^2-429812744 x+40289347\right )}{(3-2 x)^{9/2} \left (2 x^2+x+1\right )^4}}{30359089152} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((3 - 2*x)^(11/2)*(1 + x + 2*x^2)^5),x]

[Out]

((-14*(40289347 - 429812744*x + 135202154*x^2 - 1073855156*x^3 + 1627773523*x^4
- 1470758860*x^5 + 2888625656*x^6 - 3106712560*x^7 + 2343370048*x^8 - 2443779648
*x^9 + 1873554048*x^10 - 677249280*x^11 + 88070400*x^12))/((3 - 2*x)^(9/2)*(1 +
x + 2*x^2)^4) + ((45*I)*(53515*I + 284993*Sqrt[7])*ArcTan[Sqrt[6 - 4*x]/Sqrt[-7
- I*Sqrt[7]]])/Sqrt[(-I/2)*(-7*I + Sqrt[7])] - ((45*I)*(-53515*I + 284993*Sqrt[7
])*ArcTan[Sqrt[6 - 4*x]/Sqrt[-7 + I*Sqrt[7]]])/Sqrt[(I/2)*(7*I + Sqrt[7])])/3035
9089152

_______________________________________________________________________________________

Maple [A]  time = 0.132, size = 584, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3-2*x)^(11/2)/(2*x^2+x+1)^5,x)

[Out]

1/6588344*(567651623/32*(3-2*x)^(1/2)-6194606411/192*(3-2*x)^(3/2)[phone]/38
4*(3-2*x)^(5/2)-8763772549/768*(3-2*x)^(7/2)[phone]/48*(3-2*x)^(9/2)-20006363
3/384*(3-2*x)^(11/2)[phone]/384*(3-2*x)^(13/2)-526135/256*(3-2*x)^(15/2))/((3-
2*x)^2-7+14*x)^4-731595/13492928512*ln(3-2*x+14^(1/2)-(3-2*x)^(1/2)*(7+2*14^(1/2
))^(1/2))*(7+2*14^(1/2))^(1/2)*14^(1/2)+1424965/6746464256*ln(3-2*x+14^(1/2)-(3-
2*x)^(1/2)*(7+2*14^(1/2))^(1/2))*(7+2*14^(1/2))^(1/2)-731595/6746464256/(-7+2*14
^(1/2))^(1/2)*arctan((2*(3-2*x)^(1/2)-(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2
))*(7+2*14^(1/2))*14^(1/2)+1424965/3373232128/(-7+2*14^(1/2))^(1/2)*arctan((2*(3
-2*x)^(1/2)-(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*(7+2*14^(1/2))-578695/3
373232128/(-7+2*14^(1/2))^(1/2)*arctan((2*(3-2*x)^(1/2)-(7+2*14^(1/2))^(1/2))/(-
7+2*14^(1/2))^(1/2))*14^(1/2)+731595/13492928512*ln(3-2*x+14^(1/2)+(3-2*x)^(1/2)
*(7+2*14^(1/2))^(1/2))*(7+2*14^(1/2))^(1/2)*14^(1/2)-1424965/6746464256*ln(3-2*x
+14^(1/2)+(3-2*x)^(1/2)*(7+2*14^(1/2))^(1/2))*(7+2*14^(1/2))^(1/2)-731595/674646
4256/(-7+2*14^(1/2))^(1/2)*arctan((2*(3-2*x)^(1/2)+(7+2*14^(1/2))^(1/2))/(-7+2*1
4^(1/2))^(1/2))*(7+2*14^(1/2))*14^(1/2)+1424965/3373232128/(-7+2*14^(1/2))^(1/2)
*arctan((2*(3-2*x)^(1/2)+(7+2*14^(1/2))^(1/2))/(-7+2*14^(1/2))^(1/2))*(7+2*14^(1
/2))-578695/3373232128/(-7+2*14^(1/2))^(1/2)*arctan((2*(3-2*x)^(1/2)+(7+2*14^(1/
2))^(1/2))/(-7+2*14^(1/2))^(1/2))*14^(1/2)+1/151263/(3-2*x)^(9/2)+5/235298/(3-2*
x)^(7/2)+19/470596/(3-2*x)^(5/2)+185/2823576/(3-2*x)^(3/2)+505/3294172/(3-2*x)^(
1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, x^{2} + x + 1\right )}^{5}{\left (-2 \, x + 3\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*x^2 + x + 1)^5*(-2*x + 3)^(11/2)),x, algorithm="maxima")

[Out]

integrate(1/((2*x^2 + x + 1)^5*(-2*x + 3)^(11/2)), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.280812, size = 1827, normalized size = 4.49 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*x^2 + x + 1)^5*(-2*x + 3)^(11/2)),x, algorithm="fricas")

[Out]

1/948692239236095606784*sqrt(79716926)*2744^(3/4)*(2263908918780*sqrt(39858463)*
sqrt(14)*(256*x^12 - 1024*x^11 + 1280*x^10 - 1024*x^9 + 1888*x^8 - 1664*x^7 + 56
0*x^6 - 1280*x^5 + 577*x^4 + 12*x^3 + 486*x^2 + 108*x + 81)*sqrt(-2*x + 3)*arcta
n(1116036964*sqrt(39858463)*(284993*sqrt(14)*sqrt(7) - 1024233*sqrt(7))/(sqrt(79
716926)*sqrt(39858463)*2744^(1/4)*(21292357711*sqrt(14)*sqrt(7) - 81630132224*sq
rt(7))*sqrt(-sqrt(14)*(sqrt(79716926)*sqrt(39858463)*2744^(1/4)*sqrt(-2*x + 3)*(
40473233043683538146117038154862650739*sqrt(14) - 151437300139796837407365155111
661919174)*sqrt((21292357711*sqrt(14) - 81630132224)/(1738097975309635979264*sqr
t(14) - 6505290721706129709735)) + 79716926*sqrt(14)*(28039414460340853832967026
1219721*sqrt(14)*(2*x - 3) - 2098285190694476737313101842132992*x + 314742778604
1715105969652763199488) - 312930229866565049169122829452744361767044*sqrt(14) +
1170881916914412434739769848658455837958144)/(280394144603408538329670261219721*
sqrt(14) - 1049142595347238368656550921066496))*sqrt((21292357711*sqrt(14) - 816
30132224)/(1738097975309635979264*sqrt(14) - 6505290721706129709735)) + 55801848
2*sqrt(79716926)*2744^(1/4)*sqrt(-2*x + 3)*(21292357711*sqrt(14) - 81630132224)*
sqrt((21292357711*sqrt(14) - 81630132224)/(1738097975309635979264*sqrt(14) - 650
5290721706129709735)) - 7812258748*sqrt(39858463)*(7645*sqrt(14) - 115739))) + 2
263908918780*sqrt(39858463)*sqrt(14)*(256*x^12 - 1024*x^11 + 1280*x^10 - 1024*x^
9 + 1888*x^8 - 1664*x^7 + 560*x^6 - 1280*x^5 + 577*x^4 + 12*x^3 + 486*x^2 + 108*
x + 81)*sqrt(-2*x + 3)*arctan(1116036964*sqrt(39858463)*(284993*sqrt(14)*sqrt(7)
- 1024233*sqrt(7))/(sqrt(79716926)*sqrt(39858463)*2744^(1/4)*(21292357711*sqrt(
14)*sqrt(7) - 81630132224*sqrt(7))*sqrt(sqrt(14)*(sqrt(79716926)*sqrt(39858463)*
2744^(1/4)*sqrt(-2*x + 3)*(40473233043683538146117038154862650739*sqrt(14) - 151
437300139796837407365155111661919174)*sqrt((21292357711*sqrt(14) - 81630132224)/
(1738097975309635979264*sqrt(14) - 6505290721706129709735)) - 79716926*sqrt(14)*
(280394144603408538329670261219721*sqrt(14)*(2*x - 3) - 209828519069447673731310
1842132992*x + 3147427786041715105969652763199488) + 312930229866565049169122829
452744361767044*sqrt(14) - 1170881916914412434739769848658455837958144)/(2803941
44603408538329670261219721*sqrt(14) - 1049142595347238368656550921066496))*sqrt(
(21292357711*sqrt(14) - 81630132224)/(1738097975309635979264*sqrt(14) - 65052907
21706129709735)) + 558018482*sqrt(79716926)*2744^(1/4)*sqrt(-2*x + 3)*(212923577
11*sqrt(14) - 81630132224)*sqrt((21292357711*sqrt(14) - 81630132224)/(1738097975
309635979264*sqrt(14) - 6505290721706129709735)) + 7812258748*sqrt(39858463)*(76
45*sqrt(14) - 115739))) + 45*sqrt(39858463)*(21292357711*sqrt(14)*sqrt(7)*(256*x
^12 - 1024*x^11 + 1280*x^10 - 1024*x^9 + 1888*x^8 - 1664*x^7 + 560*x^6 - 1280*x^
5 + 577*x^4 + 12*x^3 + 486*x^2 + 108*x + 81) - 81630132224*sqrt(7)*(256*x^12 - 1
024*x^11 + 1280*x^10 - 1024*x^9 + 1888*x^8 - 1664*x^7 + 560*x^6 - 1280*x^5 + 577
*x^4 + 12*x^3 + 486*x^2 + 108*x + 81))*sqrt(-2*x + 3)*log(-996461575/4*sqrt(14)*
(sqrt(79716926)*sqrt(39858463)*2744^(1/4)*sqrt(-2*x + 3)*(4047323304368353814611
7038154862650739*sqrt(14) - 151437300139796837407365155111661919174)*sqrt((21292
357711*sqrt(14) - 81630132224)/(1738097975309635979264*sqrt(14) - 65052907217061
29709735)) + 79716926*sqrt(14)*(280394144603408538329670261219721*sqrt(14)*(2*x
- 3) - 2098285190694476737313101842132992*x + 3147427786041715105969652763199488
) - 312930229866565049169122829452744361767044*sqrt(14) + 1170881916914412434739
769848658455837958144)/(280394144603408538329670261219721*sqrt(14) - 10491425953
47238368656550921066496)) - 45*sqrt(39858463)*(21292357711*sqrt(14)*sqrt(7)*(256
*x^12 - 1024*x^11 + 1280*x^10 - 1024*x^9 + 1888*x^8 - 1664*x^7 + 560*x^6 - 1280*
x^5 + 577*x^4 + 12*x^3 + 486*x^2 + 108*x + 81) - 81630132224*sqrt(7)*(256*x^12 -
1024*x^11 + 1280*x^10 - 1024*x^9 + 1888*x^8 - 1664*x^7 + 560*x^6 - 1280*x^5 + 5
77*x^4 + 12*x^3 + 486*x^2 + 108*x + 81))*sqrt(-2*x + 3)*log(996461575/4*sqrt(14)
*(sqrt(79716926)*sqrt(39858463)*2744^(1/4)*sqrt(-2*x + 3)*(404732330436835381461
17038154862650739*sqrt(14) - 151437300139796837407365155111661919174)*sqrt((2129
2357711*sqrt(14) - 81630132224)/(1738097975309635979264*sqrt(14) - 6505290721706
129709735)) - 79716926*sqrt(14)*(280394144603408538329670261219721*sqrt(14)*(2*x
- 3) - 2098285190694476737313101842132992*x + 314742778604171510596965276319948
8) + 312930229866565049169122829452744361767044*sqrt(14) - 117088191691441243473
9769848658455837958144)/(280394144603408538329670261219721*sqrt(14) - 1049142595
347238368656550921066496)) - 2*sqrt(79716926)*2744^(1/4)*(21292357711*sqrt(14)*s
qrt(7)*(88070400*x^12 - 677249280*x^11 + 1873554048*x^10 - 2443779648*x^9 + 2343
370048*x^8 - 3106712560*x^7 + 2888625656*x^6 - 1470758860*x^5 + 1627773523*x^4 -
1073855156*x^3 + 135202154*x^2 - 429812744*x + 40289347) - 81630132224*sqrt(7)*
(88070400*x^12 - 677249280*x^11 + 1873554048*x^10 - 2443779648*x^9 + 2343370048*
x^8 - 3106712560*x^7 + 2888625656*x^6 - 1470758860*x^5 + 1627773523*x^4 - 107385
5156*x^3 + 135202154*x^2 - 429812744*x + 40289347))*sqrt((21292357711*sqrt(14) -
81630132224)/(1738097975309635979264*sqrt(14) - 6505290721706129709735)))/((212
92357711*sqrt(14)*sqrt(7)*(256*x^12 - 1024*x^11 + 1280*x^10 - 1024*x^9 + 1888*x^
8 - 1664*x^7 + 560*x^6 - 1280*x^5 + 577*x^4 + 12*x^3 + 486*x^2 + 108*x + 81) - 8
1630132224*sqrt(7)*(256*x^12 - 1024*x^11 + 1280*x^10 - 1024*x^9 + 1888*x^8 - 166
4*x^7 + 560*x^6 - 1280*x^5 + 577*x^4 + 12*x^3 + 486*x^2 + 108*x + 81))*sqrt(-2*x
+ 3)*sqrt((21292357711*sqrt(14) - 81630132224)/(1738097975309635979264*sqrt(14)
- 6505290721706129709735)))

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3-2*x)**(11/2)/(2*x**2+x+1)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, x^{2} + x + 1\right )}^{5}{\left (-2 \, x + 3\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*x^2 + x + 1)^5*(-2*x + 3)^(11/2)),x, algorithm="giac")

[Out]

integrate(1/((2*x^2 + x + 1)^5*(-2*x + 3)^(11/2)), x)