3.2318 \(\int \frac{1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx\)
Optimal. Leaf size=313 \[ \frac{20 x+37}{434 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )^2}+\frac{5 (2080 x+2329)}{94178 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )}-\frac{81090}{329623 \sqrt{2 x+1}}-\frac{15 \sqrt{\frac{1}{434} \left (2257111762+387427075 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{659246}+\frac{15 \sqrt{\frac{1}{434} \left (2257111762+387427075 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{659246}-\frac{15 \sqrt{\frac{1}{434} \left (387427075 \sqrt{35}-2257111762\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{329623}+\frac{15 \sqrt{\frac{1}{434} \left (387427075 \sqrt{35}-2257111762\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{329623} \]
[Out]
-81090/(329623*Sqrt[1 + 2*x]) + (37 + 20*x)/(434*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)
^2) + (5*(2329 + 2080*x))/(94178*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) - (15*Sqrt[(-2
257111762 + 387427075*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1
+ 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/329623 + (15*Sqrt[(-2257111762 + 387427075*S
qrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 +
Sqrt[35])]])/329623 - (15*Sqrt[(2257111762 + 387427075*Sqrt[35])/434]*Log[Sqrt[3
5] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/659246 + (15*Sqrt[(22
57111762 + 387427075*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[
1 + 2*x] + 5*(1 + 2*x)])/659246
_______________________________________________________________________________________
Rubi [A] time = 1.30875, antiderivative size = 313, normalized size of antiderivative = 1.,
number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409
\[ \frac{20 x+37}{434 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )^2}+\frac{5 (2080 x+2329)}{94178 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )}-\frac{81090}{329623 \sqrt{2 x+1}}-\frac{15 \sqrt{\frac{1}{434} \left (2257111762+387427075 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{659246}+\frac{15 \sqrt{\frac{1}{434} \left (2257111762+387427075 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{659246}-\frac{15 \sqrt{\frac{1}{434} \left (387427075 \sqrt{35}-2257111762\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{329623}+\frac{15 \sqrt{\frac{1}{434} \left (387427075 \sqrt{35}-2257111762\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{329623} \]
Antiderivative was successfully verified.
[In] Int[1/((1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)^3),x]
[Out]
-81090/(329623*Sqrt[1 + 2*x]) + (37 + 20*x)/(434*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)
^2) + (5*(2329 + 2080*x))/(94178*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) - (15*Sqrt[(-2
257111762 + 387427075*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1
+ 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/329623 + (15*Sqrt[(-2257111762 + 387427075*S
qrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 +
Sqrt[35])]])/329623 - (15*Sqrt[(2257111762 + 387427075*Sqrt[35])/434]*Log[Sqrt[3
5] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/659246 + (15*Sqrt[(22
57111762 + 387427075*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[
1 + 2*x] + 5*(1 + 2*x)])/659246
_______________________________________________________________________________________
Rubi in Sympy [A] time = 84.5443, size = 410, normalized size = 1.31 \[ - \frac{\sqrt{14} \left (40545 \sqrt{35} + 271380\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{9229444 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (40545 \sqrt{35} + 271380\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{9229444 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (81090 \sqrt{35} + 542760\right )}{10} + 108552 \sqrt{10} \sqrt{2 + \sqrt{35}}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{4614722 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (81090 \sqrt{35} + 542760\right )}{10} + 108552 \sqrt{10} \sqrt{2 + \sqrt{35}}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{4614722 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{20 x + 37}{434 \sqrt{2 x + 1} \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{10400 x + 11645}{94178 \sqrt{2 x + 1} \left (5 x^{2} + 3 x + 2\right )} - \frac{81090}{329623 \sqrt{2 x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1+2*x)**(3/2)/(5*x**2+3*x+2)**3,x)
[Out]
-sqrt(14)*(40545*sqrt(35) + 271380)*log(2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2
*x + 1)/5 + 1 + sqrt(35)/5)/(9229444*sqrt(2 + sqrt(35))) + sqrt(14)*(40545*sqrt(
35) + 271380)*log(2*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(3
5)/5)/(9229444*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(810
90*sqrt(35) + 542760)/10 + 108552*sqrt(10)*sqrt(2 + sqrt(35)))*atan(sqrt(10)*(sq
rt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(4614722*sqrt(-2 +
sqrt(35))*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)*sqrt(2 + sqrt(35))*(81090*s
qrt(35) + 542760)/10 + 108552*sqrt(10)*sqrt(2 + sqrt(35)))*atan(sqrt(10)*(sqrt(2
*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(4614722*sqrt(-2 + sqr
t(35))*sqrt(2 + sqrt(35))) + (20*x + 37)/(434*sqrt(2*x + 1)*(5*x**2 + 3*x + 2)**
2) + (10400*x + 11645)/(94178*sqrt(2*x + 1)*(5*x**2 + 3*x + 2)) - 81090/(329623*
sqrt(2*x + 1))
_______________________________________________________________________________________
Mathematica [C] time = 1.37161, size = 170, normalized size = 0.54 \[ \frac{-\frac{31 \left (4054500 x^4+4501400 x^3+4077245 x^2+1525635 x+429487\right )}{2 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )^2}-\frac{15 i \left (12686 \sqrt{31}-83793 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{\sqrt{-\frac{1}{5} i \left (\sqrt{31}-2 i\right )}}+\frac{15 i \left (12686 \sqrt{31}+83793 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{\sqrt{\frac{1}{5} i \left (\sqrt{31}+2 i\right )}}}{10218313} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)^3),x]
[Out]
((-31*(429487 + 1525635*x + 4077245*x^2 + 4501400*x^3 + 4054500*x^4))/(2*Sqrt[1
+ 2*x]*(2 + 3*x + 5*x^2)^2) - ((15*I)*(-83793*I + 12686*Sqrt[31])*ArcTan[Sqrt[5
+ 10*x]/Sqrt[-2 - I*Sqrt[31]]])/Sqrt[(-I/5)*(-2*I + Sqrt[31])] + ((15*I)*(83793*
I + 12686*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]])/Sqrt[(I/5)*(2*
I + Sqrt[31])])/10218313
_______________________________________________________________________________________
Maple [B] time = 0.064, size = 671, normalized size = 2.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1+2*x)^(3/2)/(5*x^2+3*x+2)^3,x)
[Out]
-64/343/(1+2*x)^(1/2)-1600/343*(9793/30752*(1+2*x)^(7/2)-14343/19220*(1+2*x)^(5/
2)+762223/768800*(1+2*x)^(3/2)-170877/192200*(1+2*x)^(1/2))/(5*(1+2*x)^2-8*x+3)^
2-95145/20436626*ln(-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5^(1/2)*7
^(1/2)+10*x+5)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-876315/286112764*ln(-(2*5^(1/
2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5^(1/2)*7^(1/2)+10*x+5)*7^(1/2)*(2*5^(
1/2)*7^(1/2)+4)^(1/2)-475725/10218313/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-(2*
5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)+10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2)
)*(2*5^(1/2)*7^(1/2)+4)-876315/143056382/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-
(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)+10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1
/2))*(2*5^(1/2)*7^(1/2)+4)*5^(1/2)*7^(1/2)+542760/2307361/(10*5^(1/2)*7^(1/2)-20
)^(1/2)*arctan((-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)+10*(1+2*x)^(1/2))/(10*5^(1/
2)*7^(1/2)-20)^(1/2))*5^(1/2)*7^(1/2)+95145/20436626*ln(5^(1/2)*7^(1/2)+10*x+5+(
2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^
(1/2)+876315/286112764*ln(5^(1/2)*7^(1/2)+10*x+5+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(
1/2)*(1+2*x)^(1/2))*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-475725/10218313/(10*5^(1
/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/
2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)-876315/143056382/(10*5^
(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(
1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)*5^(1/2)*7^(1/2)+54276
0/2307361/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)+(2*5^(1/2)*7^(1
/2)+4)^(1/2)*5^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*5^(1/2)*7^(1/2)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}{\left (2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x^2 + 3*x + 2)^3*(2*x + 1)^(3/2)),x, algorithm="maxima")
[Out]
integrate(1/((5*x^2 + 3*x + 2)^3*(2*x + 1)^(3/2)), x)
_______________________________________________________________________________________
Fricas [A] time = 0.292385, size = 1485, normalized size = 4.74 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x^2 + 3*x + 2)^3*(2*x + 1)^(3/2)),x, algorithm="fricas")
[Out]
-1/4260990634275782932*sqrt(632534)*329623^(3/4)*sqrt(31)*(932272999980*10206613
805^(1/4)*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(2*x + 1)*arctan(480
409573*10206613805^(1/4)*sqrt(31)*(63430*sqrt(7) + 58421*sqrt(5))/(sqrt(7003055)
*sqrt(632534)*329623^(1/4)*sqrt(31)*(2257111762*sqrt(7) + 2711989525*sqrt(5))*sq
rt(sqrt(7)*(10206613805^(1/4)*sqrt(632534)*329623^(1/4)*(10934254835900318812488
241206938582472922910445653165861359814285844308640325338579111840534960936097*s
qrt(7)*sqrt(5) + 646879239779290859249970060739633876754449409479123455803473934
41202599599861940881783625777173580608)*sqrt(2*x + 1)*sqrt((2257111762*sqrt(7)*s
qrt(5) + 13559947625)/(1748932415799512300*sqrt(7)*sqrt(5) + 1034804435166569151
9)) + 1581335*sqrt(7)*(320811029955767011032857681971637828573623492460506264327
044549226802932959543432507839543241500*sqrt(7)*sqrt(5)*(2*x + 1) + 379588729703
33570468973585088477327993850641657844517404031996309530077522909943012800647566
67198*x + 1897943648516678523448679254423866399692532082892225870201599815476503
876145497150640032378333599) + 2213869*sqrt(5)*(32081102995576701103285768197163
7828573623492460506264327044549226802932959543432507839543241500*sqrt(7)*sqrt(5)
+ 18979436485166785234486792544238663996925320828922258702015998154765038761454
97150640032378333599))/(32081102995576701103285768197163782857362349246050626432
7044549226802932959543432507839543241500*sqrt(7)*sqrt(5) + 189794364851667852344
8679254423866399692532082892225870201599815476503876145497150640032378333599))*s
qrt((2257111762*sqrt(7)*sqrt(5) + 13559947625)/(1748932415799512300*sqrt(7)*sqrt
(5) + 10348044351665691519)) + 49021385*sqrt(632534)*329623^(1/4)*sqrt(2*x + 1)*
(2257111762*sqrt(7) + 2711989525*sqrt(5))*sqrt((2257111762*sqrt(7)*sqrt(5) + 135
59947625)/(1748932415799512300*sqrt(7)*sqrt(5) + 10348044351665691519)) + 148926
96763*10206613805^(1/4)*(13515*sqrt(7) + 18092*sqrt(5)))) + 932272999980*1020661
3805^(1/4)*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(2*x + 1)*arctan(48
0409573*10206613805^(1/4)*sqrt(31)*(63430*sqrt(7) + 58421*sqrt(5))/(sqrt(7003055
)*sqrt(632534)*329623^(1/4)*sqrt(31)*(2257111762*sqrt(7) + 2711989525*sqrt(5))*s
qrt(-sqrt(7)*(10206613805^(1/4)*sqrt(632534)*329623^(1/4)*(109342548359003188124
88241206938582472922910445653165861359814285844308640325338579111840534960936097
*sqrt(7)*sqrt(5) + 6468792397792908592499700607396338767544494094791234558034739
3441202599599861940881783625777173580608)*sqrt(2*x + 1)*sqrt((2257111762*sqrt(7)
*sqrt(5) + 13559947625)/(1748932415799512300*sqrt(7)*sqrt(5) + 10348044351665691
519)) - 1581335*sqrt(7)*(3208110299557670110328576819716378285736234924605062643
27044549226802932959543432507839543241500*sqrt(7)*sqrt(5)*(2*x + 1) + 3795887297
03335704689735850884773279938506416578445174040319963095300775229099430128006475
6667198*x + 18979436485166785234486792544238663996925320828922258702015998154765
03876145497150640032378333599) - 2213869*sqrt(5)*(320811029955767011032857681971
637828573623492460506264327044549226802932959543432507839543241500*sqrt(7)*sqrt(
5) + 189794364851667852344867925442386639969253208289222587020159981547650387614
5497150640032378333599))/(320811029955767011032857681971637828573623492460506264
327044549226802932959543432507839543241500*sqrt(7)*sqrt(5) + 1897943648516678523
448679254423866399692532082892225870201599815476503876145497150640032378333599))
*sqrt((2257111762*sqrt(7)*sqrt(5) + 13559947625)/(1748932415799512300*sqrt(7)*sq
rt(5) + 10348044351665691519)) + 49021385*sqrt(632534)*329623^(1/4)*sqrt(2*x + 1
)*(2257111762*sqrt(7) + 2711989525*sqrt(5))*sqrt((2257111762*sqrt(7)*sqrt(5) + 1
3559947625)/(1748932415799512300*sqrt(7)*sqrt(5) + 10348044351665691519)) - 1489
2696763*10206613805^(1/4)*(13515*sqrt(7) + 18092*sqrt(5)))) - 105*10206613805^(1
/4)*sqrt(31)*(2257111762*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 2711989
525*sqrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*sqrt(2*x + 1)*log(98480460937
5/49*sqrt(7)*(10206613805^(1/4)*sqrt(632534)*329623^(1/4)*(109342548359003188124
88241206938582472922910445653165861359814285844308640325338579111840534960936097
*sqrt(7)*sqrt(5) + 6468792397792908592499700607396338767544494094791234558034739
3441202599599861940881783625777173580608)*sqrt(2*x + 1)*sqrt((2257111762*sqrt(7)
*sqrt(5) + 13559947625)/(1748932415799512300*sqrt(7)*sqrt(5) + 10348044351665691
519)) + 1581335*sqrt(7)*(3208110299557670110328576819716378285736234924605062643
27044549226802932959543432507839543241500*sqrt(7)*sqrt(5)*(2*x + 1) + 3795887297
03335704689735850884773279938506416578445174040319963095300775229099430128006475
6667198*x + 18979436485166785234486792544238663996925320828922258702015998154765
03876145497150640032378333599) + 2213869*sqrt(5)*(320811029955767011032857681971
637828573623492460506264327044549226802932959543432507839543241500*sqrt(7)*sqrt(
5) + 189794364851667852344867925442386639969253208289222587020159981547650387614
5497150640032378333599))/(320811029955767011032857681971637828573623492460506264
327044549226802932959543432507839543241500*sqrt(7)*sqrt(5) + 1897943648516678523
448679254423866399692532082892225870201599815476503876145497150640032378333599))
+ 105*10206613805^(1/4)*sqrt(31)*(2257111762*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2
+ 12*x + 4) + 2711989525*sqrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*sqrt(2*x
+ 1)*log(-984804609375/49*sqrt(7)*(10206613805^(1/4)*sqrt(632534)*329623^(1/4)*
(1093425483590031881248824120693858247292291044565316586135981428584430864032533
8579111840534960936097*sqrt(7)*sqrt(5) + 646879239779290859249970060739633876754
44940947912345580347393441202599599861940881783625777173580608)*sqrt(2*x + 1)*sq
rt((2257111762*sqrt(7)*sqrt(5) + 13559947625)/(1748932415799512300*sqrt(7)*sqrt(
5) + 10348044351665691519)) - 1581335*sqrt(7)*(320811029955767011032857681971637
828573623492460506264327044549226802932959543432507839543241500*sqrt(7)*sqrt(5)*
(2*x + 1) + 37958872970333570468973585088477327993850641657844517404031996309530
07752290994301280064756667198*x + 1897943648516678523448679254423866399692532082
892225870201599815476503876145497150640032378333599) - 2213869*sqrt(5)*(32081102
99557670110328576819716378285736234924605062643270445492268029329595434325078395
43241500*sqrt(7)*sqrt(5) + 18979436485166785234486792544238663996925320828922258
70201599815476503876145497150640032378333599))/(32081102995576701103285768197163
7828573623492460506264327044549226802932959543432507839543241500*sqrt(7)*sqrt(5)
+ 18979436485166785234486792544238663996925320828922258702015998154765038761454
97150640032378333599)) + sqrt(632534)*329623^(1/4)*sqrt(31)*(2257111762*sqrt(7)*
(4054500*x^4 + 4501400*x^3 + 4077245*x^2 + 1525635*x + 429487) + 2711989525*sqrt
(5)*(4054500*x^4 + 4501400*x^3 + 4077245*x^2 + 1525635*x + 429487))*sqrt((225711
1762*sqrt(7)*sqrt(5) + 13559947625)/(1748932415799512300*sqrt(7)*sqrt(5) + 10348
044351665691519)))/((2257111762*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) +
2711989525*sqrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*sqrt(2*x + 1)*sqrt((22
57111762*sqrt(7)*sqrt(5) + 13559947625)/(1748932415799512300*sqrt(7)*sqrt(5) + 1
0348044351665691519)))
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (2 x + 1\right )^{\frac{3}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1+2*x)**(3/2)/(5*x**2+3*x+2)**3,x)
[Out]
Integral(1/((2*x + 1)**(3/2)*(5*x**2 + 3*x + 2)**3), x)
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}{\left (2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x^2 + 3*x + 2)^3*(2*x + 1)^(3/2)),x, algorithm="giac")
[Out]
integrate(1/((5*x^2 + 3*x + 2)^3*(2*x + 1)^(3/2)), x)