3.2317 \(\int \frac{1}{\sqrt{1+2 x} \left (2+3 x+5 x^2\right )^3} \, dx\)
Optimal. Leaf size=314 \[ \frac{\sqrt{2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}+\frac{\sqrt{2 x+1} (7920 x+9227)}{94178 \left (5 x^2+3 x+2\right )}-\frac{3 \sqrt{\frac{1}{434} \left (64681225 \sqrt{35}-250141922\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{94178}+\frac{3 \sqrt{\frac{1}{434} \left (64681225 \sqrt{35}-250141922\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{94178}-\frac{3 \sqrt{\frac{1}{434} \left (2+\sqrt{35}\right )} \left (7379+264 \sqrt{35}\right ) \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{47089}+\frac{3 \sqrt{\frac{1}{434} \left (2+\sqrt{35}\right )} \left (7379+264 \sqrt{35}\right ) \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{47089} \]
[Out]
(Sqrt[1 + 2*x]*(37 + 20*x))/(434*(2 + 3*x + 5*x^2)^2) + (Sqrt[1 + 2*x]*(9227 + 7
920*x))/(94178*(2 + 3*x + 5*x^2)) - (3*Sqrt[(2 + Sqrt[35])/434]*(7379 + 264*Sqrt
[35])*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35]
)]])/47089 + (3*Sqrt[(2 + Sqrt[35])/434]*(7379 + 264*Sqrt[35])*ArcTan[(Sqrt[10*(
2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/47089 - (3*Sqrt[(-
250141922 + 64681225*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[
1 + 2*x] + 5*(1 + 2*x)])/94178 + (3*Sqrt[(-250141922 + 64681225*Sqrt[35])/434]*L
og[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/94178
_______________________________________________________________________________________
Rubi [A] time = 1.27574, antiderivative size = 314, normalized size of antiderivative = 1.,
number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364
\[ \frac{\sqrt{2 x+1} (20 x+37)}{434 \left (5 x^2+3 x+2\right )^2}+\frac{\sqrt{2 x+1} (7920 x+9227)}{94178 \left (5 x^2+3 x+2\right )}-\frac{3 \sqrt{\frac{1}{434} \left (64681225 \sqrt{35}-250141922\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{94178}+\frac{3 \sqrt{\frac{1}{434} \left (64681225 \sqrt{35}-250141922\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{94178}-\frac{3 \sqrt{\frac{1}{434} \left (2+\sqrt{35}\right )} \left (7379+264 \sqrt{35}\right ) \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{47089}+\frac{3 \sqrt{\frac{1}{434} \left (2+\sqrt{35}\right )} \left (7379+264 \sqrt{35}\right ) \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{47089} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^3),x]
[Out]
(Sqrt[1 + 2*x]*(37 + 20*x))/(434*(2 + 3*x + 5*x^2)^2) + (Sqrt[1 + 2*x]*(9227 + 7
920*x))/(94178*(2 + 3*x + 5*x^2)) - (3*Sqrt[(2 + Sqrt[35])/434]*(7379 + 264*Sqrt
[35])*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35]
)]])/47089 + (3*Sqrt[(2 + Sqrt[35])/434]*(7379 + 264*Sqrt[35])*ArcTan[(Sqrt[10*(
2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/47089 - (3*Sqrt[(-
250141922 + 64681225*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[
1 + 2*x] + 5*(1 + 2*x)])/94178 + (3*Sqrt[(-250141922 + 64681225*Sqrt[35])/434]*L
og[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/94178
_______________________________________________________________________________________
Rubi in Sympy [A] time = 77.6422, size = 400, normalized size = 1.27 \[ \frac{\sqrt{2 x + 1} \left (20 x + 37\right )}{434 \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{\sqrt{2 x + 1} \left (7920 x + 9227\right )}{94178 \left (5 x^{2} + 3 x + 2\right )} - \frac{\sqrt{14} \left (- 792 \sqrt{35} + 22137\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{1318492 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (- 792 \sqrt{35} + 22137\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{1318492 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 1584 \sqrt{35} + 44274\right )}{10} + \frac{44274 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{659246 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 1584 \sqrt{35} + 44274\right )}{10} + \frac{44274 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{659246 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1+2*x)**(1/2)/(5*x**2+3*x+2)**3,x)
[Out]
sqrt(2*x + 1)*(20*x + 37)/(434*(5*x**2 + 3*x + 2)**2) + sqrt(2*x + 1)*(7920*x +
9227)/(94178*(5*x**2 + 3*x + 2)) - sqrt(14)*(-792*sqrt(35) + 22137)*log(2*x - sq
rt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(1318492*sqrt(2 + sq
rt(35))) + sqrt(14)*(-792*sqrt(35) + 22137)*log(2*x + sqrt(10)*sqrt(2 + sqrt(35)
)*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(1318492*sqrt(2 + sqrt(35))) + sqrt(35)*(-sq
rt(10)*sqrt(2 + sqrt(35))*(-1584*sqrt(35) + 44274)/10 + 44274*sqrt(10)*sqrt(2 +
sqrt(35))/5)*atan(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 +
sqrt(35)))/(659246*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10
)*sqrt(2 + sqrt(35))*(-1584*sqrt(35) + 44274)/10 + 44274*sqrt(10)*sqrt(2 + sqrt(
35))/5)*atan(sqrt(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt
(35)))/(659246*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35)))
_______________________________________________________________________________________
Mathematica [C] time = 1.20701, size = 161, normalized size = 0.51 \[ \frac{\frac{31 \sqrt{2 x+1} \left (39600 x^3+69895 x^2+47861 x+26483\right )}{2 \left (5 x^2+3 x+2\right )^2}+\frac{3 \left (8184-7907 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{\sqrt{-\frac{1}{5} i \left (\sqrt{31}-2 i\right )}}+\frac{3 \left (8184+7907 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{\sqrt{\frac{1}{5} i \left (\sqrt{31}+2 i\right )}}}{1459759} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^3),x]
[Out]
((31*Sqrt[1 + 2*x]*(26483 + 47861*x + 69895*x^2 + 39600*x^3))/(2*(2 + 3*x + 5*x^
2)^2) + (3*(8184 - (7907*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]
]])/Sqrt[(-I/5)*(-2*I + Sqrt[31])] + (3*(8184 + (7907*I)*Sqrt[31])*ArcTan[Sqrt[5
+ 10*x]/Sqrt[-2 + I*Sqrt[31]]])/Sqrt[(I/5)*(2*I + Sqrt[31])])/1459759
_______________________________________________________________________________________
Maple [B] time = 0.542, size = 1398, normalized size = 4.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x)
[Out]
-5/20436626*(-6/1353025*(-6045943503600+620096769600*5^(1/2)*7^(1/2))/(-390+40*5
^(1/2)*7^(1/2))*(1+2*x)^(3/2)+1/6765125/(-390+40*5^(1/2)*7^(1/2))*(-910485268182
00*5^(1/2)+65791327714000*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(1+2*x)-2/6765125
*(-59423591568600*5^(1/2)*7^(1/2)+320925328420550)/(-390+40*5^(1/2)*7^(1/2))*(1+
2*x)^(1/2)+1/13530250*(-123371070933600*7^(1/2)+152992435939000*5^(1/2))*(2*5^(1
/2)*7^(1/2)+4)^(1/2)/(-390+40*5^(1/2)*7^(1/2)))/(1/5*5^(1/2)*7^(1/2)+2*x+1-1/5*(
2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))^2-30327375/5839036/(-100*5^(1/
2)*7^(1/2)+975)*ln(5*(4*5^(1/2)*7^(1/2)-39)*((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)+7660
8825/20436626/(-100*5^(1/2)*7^(1/2)+975)*ln(5*(4*5^(1/2)*7^(1/2)-39)*((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)+1726686/329623/(27050*5^(1/2)*7^(1/2)-150820)^(1/2)*arctan(
1/5*(2*(-100*5^(1/2)*7^(1/2)+975)*(1+2*x)^(1/2)+5*(4*5^(1/2)*7^(1/2)-39)*(2*5^(1
/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(27050*5^(1/2)*7^(1/2)-150820)^(1/2))*5^(1/2)*7^(1
/2)-885480/47089/(27050*5^(1/2)*7^(1/2)-150820)^(1/2)*arctan(1/5*(2*(-100*5^(1/2
)*7^(1/2)+975)*(1+2*x)^(1/2)+5*(4*5^(1/2)*7^(1/2)-39)*(2*5^(1/2)*7^(1/2)+4)^(1/2
)*5^(1/2))/(27050*5^(1/2)*7^(1/2)-150820)^(1/2))+5110660425/10218313/(27050*5^(1
/2)*7^(1/2)-150820)^(1/2)*arctan(1/5*(2*(-100*5^(1/2)*7^(1/2)+975)*(1+2*x)^(1/2)
+5*(4*5^(1/2)*7^(1/2)-39)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(27050*5^(1/2)*7^
(1/2)-150820)^(1/2))*(2*5^(1/2)*7^(1/2)+4)*5^(1/2)/(-100*5^(1/2)*7^(1/2)+975)*7^
(1/2)-8978191125/2919518/(27050*5^(1/2)*7^(1/2)-150820)^(1/2)*arctan(1/5*(2*(-10
0*5^(1/2)*7^(1/2)+975)*(1+2*x)^(1/2)+5*(4*5^(1/2)*7^(1/2)-39)*(2*5^(1/2)*7^(1/2)
+4)^(1/2)*5^(1/2))/(27050*5^(1/2)*7^(1/2)-150820)^(1/2))*(2*5^(1/2)*7^(1/2)+4)/(
-100*5^(1/2)*7^(1/2)+975)+5/20436626*(6/1353025*(-6045943503600+620096769600*5^(
1/2)*7^(1/2))/(-390+40*5^(1/2)*7^(1/2))*(1+2*x)^(3/2)+1/6765125/(-390+40*5^(1/2)
*7^(1/2))*(-91048526818200*5^(1/2)+65791327714000*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)
^(1/2)*(1+2*x)+2/6765125*(-59423591568600*5^(1/2)*7^(1/2)+320925328420550)/(-390
+40*5^(1/2)*7^(1/2))*(1+2*x)^(1/2)+1/13530250*(-123371070933600*7^(1/2)+15299243
5939000*5^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)/(-390+40*5^(1/2)*7^(1/2)))/(1/5*5^(
1/2)*7^(1/2)+2*x+1+1/5*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))^2+3032
7375/5839036/(-100*5^(1/2)*7^(1/2)+975)*ln(-5*(4*5^(1/2)*7^(1/2)-39)*(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)-76608825/20436626/(-100*5^(1/2)*7^(1/2)+975)*ln(-5*(4*5^(1/
2)*7^(1/2)-39)*(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)+1726686/329623/(27050*5^(1/2)*7^(
1/2)-150820)^(1/2)*arctan(1/5*(2*(-100*5^(1/2)*7^(1/2)+975)*(1+2*x)^(1/2)-5*(4*5
^(1/2)*7^(1/2)-39)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(27050*5^(1/2)*7^(1/2)-1
50820)^(1/2))*5^(1/2)*7^(1/2)-885480/47089/(27050*5^(1/2)*7^(1/2)-150820)^(1/2)*
arctan(1/5*(2*(-100*5^(1/2)*7^(1/2)+975)*(1+2*x)^(1/2)-5*(4*5^(1/2)*7^(1/2)-39)*
(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(27050*5^(1/2)*7^(1/2)-150820)^(1/2))+51106
60425/10218313/(27050*5^(1/2)*7^(1/2)-150820)^(1/2)*arctan(1/5*(2*(-100*5^(1/2)*
7^(1/2)+975)*(1+2*x)^(1/2)-5*(4*5^(1/2)*7^(1/2)-39)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*
5^(1/2))/(27050*5^(1/2)*7^(1/2)-150820)^(1/2))*(2*5^(1/2)*7^(1/2)+4)*5^(1/2)/(-1
00*5^(1/2)*7^(1/2)+975)*7^(1/2)-8978191125/2919518/(27050*5^(1/2)*7^(1/2)-150820
)^(1/2)*arctan(1/5*(2*(-100*5^(1/2)*7^(1/2)+975)*(1+2*x)^(1/2)-5*(4*5^(1/2)*7^(1
/2)-39)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(27050*5^(1/2)*7^(1/2)-150820)^(1/2
))*(2*5^(1/2)*7^(1/2)+4)/(-100*5^(1/2)*7^(1/2)+975)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3} \sqrt{2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x^2 + 3*x + 2)^3*sqrt(2*x + 1)),x, algorithm="maxima")
[Out]
integrate(1/((5*x^2 + 3*x + 2)^3*sqrt(2*x + 1)), x)
_______________________________________________________________________________________
Fricas [A] time = 0.283761, size = 1434, normalized size = 4.57 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x^2 + 3*x + 2)^3*sqrt(2*x + 1)),x, algorithm="fricas")
[Out]
1/101625058429561828*329623^(3/4)*sqrt(105602)*sqrt(31)*(329623^(1/4)*sqrt(10560
2)*sqrt(31)*(250141922*sqrt(7)*(39600*x^3 + 69895*x^2 + 47861*x + 26483) - 45276
8575*sqrt(5)*(39600*x^3 + 69895*x^2 + 47861*x + 26483))*sqrt(2*x + 1)*sqrt((2501
41922*sqrt(7)*sqrt(5) - 2263842875)/(32358971877628900*sqrt(7)*sqrt(5) - 2089991
11504375959)) - 135434771724*284484245^(1/4)*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 +
12*x + 4)*arctan(80204719*284484245^(1/4)*sqrt(31)*(39535*sqrt(7) - 23998*sqrt(
5))/(sqrt(1169165)*329623^(1/4)*sqrt(105602)*sqrt(31)*(250141922*sqrt(7) - 45276
8575*sqrt(5))*sqrt(sqrt(7)*(284484245^(1/4)*329623^(1/4)*sqrt(105602)*(778014178
36159032480812650756748498623166306565141582047988147498195351335646277072050284
36*sqrt(7)*sqrt(5) - 46027949230603829019315248495939203970818237008559845609636
245010906105219935562476310292821)*sqrt(2*x + 1)*sqrt((250141922*sqrt(7)*sqrt(5)
- 2263842875)/(32358971877628900*sqrt(7)*sqrt(5) - 208999111504375959)) + 26400
5*sqrt(7)*(870179640529577973372080596728111645575197008532950901616287851935257
530573818652464500*sqrt(7)*sqrt(5)*(2*x + 1) - 102961076980920256255203208516523
79012251908564769041680092626442342953147427423262647598*x - 5148053849046012812
760160425826189506125954282384520840046313221171476573713711631323799) + 369607*
sqrt(5)*(87017964052957797337208059672811164557519700853295090161628785193525753
0573818652464500*sqrt(7)*sqrt(5) - 514805384904601281276016042582618950612595428
2384520840046313221171476573713711631323799))/(870179640529577973372080596728111
645575197008532950901616287851935257530573818652464500*sqrt(7)*sqrt(5) - 5148053
84904601281276016042582618950612595428238452084004631322117147657371371163132379
9))*sqrt((250141922*sqrt(7)*sqrt(5) - 2263842875)/(32358971877628900*sqrt(7)*sqr
t(5) - 208999111504375959)) + 8184155*329623^(1/4)*sqrt(105602)*sqrt(2*x + 1)*(2
50141922*sqrt(7) - 452768575*sqrt(5))*sqrt((250141922*sqrt(7)*sqrt(5) - 22638428
75)/(32358971877628900*sqrt(7)*sqrt(5) - 208999111504375959)) + 2486346289*28448
4245^(1/4)*(1320*sqrt(7) - 7379*sqrt(5)))) - 135434771724*284484245^(1/4)*sqrt(7
)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(80204719*284484245^(1/4)*sqrt(31)
*(39535*sqrt(7) - 23998*sqrt(5))/(sqrt(1169165)*329623^(1/4)*sqrt(105602)*sqrt(3
1)*(250141922*sqrt(7) - 452768575*sqrt(5))*sqrt(-sqrt(7)*(284484245^(1/4)*329623
^(1/4)*sqrt(105602)*(77801417836159032480812650756748498623166306565141582047988
14749819535133564627707205028436*sqrt(7)*sqrt(5) - 46027949230603829019315248495
939203970818237008559845609636245010906105219935562476310292821)*sqrt(2*x + 1)*s
qrt((250141922*sqrt(7)*sqrt(5) - 2263842875)/(32358971877628900*sqrt(7)*sqrt(5)
- 208999111504375959)) - 264005*sqrt(7)*(870179640529577973372080596728111645575
197008532950901616287851935257530573818652464500*sqrt(7)*sqrt(5)*(2*x + 1) - 102
96107698092025625520320851652379012251908564769041680092626442342953147427423262
647598*x - 514805384904601281276016042582618950612595428238452084004631322117147
6573713711631323799) - 369607*sqrt(5)*(87017964052957797337208059672811164557519
7008532950901616287851935257530573818652464500*sqrt(7)*sqrt(5) - 514805384904601
2812760160425826189506125954282384520840046313221171476573713711631323799))/(870
17964052957797337208059672811164557519700853295090161628785193525753057381865246
4500*sqrt(7)*sqrt(5) - 514805384904601281276016042582618950612595428238452084004
6313221171476573713711631323799))*sqrt((250141922*sqrt(7)*sqrt(5) - 2263842875)/
(32358971877628900*sqrt(7)*sqrt(5) - 208999111504375959)) + 8184155*329623^(1/4)
*sqrt(105602)*sqrt(2*x + 1)*(250141922*sqrt(7) - 452768575*sqrt(5))*sqrt((250141
922*sqrt(7)*sqrt(5) - 2263842875)/(32358971877628900*sqrt(7)*sqrt(5) - 208999111
504375959)) - 2486346289*284484245^(1/4)*(1320*sqrt(7) - 7379*sqrt(5)))) + 21*28
4484245^(1/4)*sqrt(31)*(250141922*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
- 452768575*sqrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*log(6576553125*sqrt(7
)*(284484245^(1/4)*329623^(1/4)*sqrt(105602)*(7780141783615903248081265075674849
862316630656514158204798814749819535133564627707205028436*sqrt(7)*sqrt(5) - 4602
79492306038290193152484959392039708182370085598456096362450109061052199355624763
10292821)*sqrt(2*x + 1)*sqrt((250141922*sqrt(7)*sqrt(5) - 2263842875)/(323589718
77628900*sqrt(7)*sqrt(5) - 208999111504375959)) + 264005*sqrt(7)*(87017964052957
7973372080596728111645575197008532950901616287851935257530573818652464500*sqrt(7
)*sqrt(5)*(2*x + 1) - 1029610769809202562552032085165237901225190856476904168009
2626442342953147427423262647598*x - 51480538490460128127601604258261895061259542
82384520840046313221171476573713711631323799) + 369607*sqrt(5)*(8701796405295779
73372080596728111645575197008532950901616287851935257530573818652464500*sqrt(7)*
sqrt(5) - 5148053849046012812760160425826189506125954282384520840046313221171476
573713711631323799))/(8701796405295779733720805967281116455751970085329509016162
87851935257530573818652464500*sqrt(7)*sqrt(5) - 51480538490460128127601604258261
89506125954282384520840046313221171476573713711631323799)) - 21*284484245^(1/4)*
sqrt(31)*(250141922*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 452768575*sq
rt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*log(-6576553125*sqrt(7)*(284484245^
(1/4)*329623^(1/4)*sqrt(105602)*(77801417836159032480812650756748498623166306565
14158204798814749819535133564627707205028436*sqrt(7)*sqrt(5) - 46027949230603829
019315248495939203970818237008559845609636245010906105219935562476310292821)*sqr
t(2*x + 1)*sqrt((250141922*sqrt(7)*sqrt(5) - 2263842875)/(32358971877628900*sqrt
(7)*sqrt(5) - 208999111504375959)) - 264005*sqrt(7)*(870179640529577973372080596
728111645575197008532950901616287851935257530573818652464500*sqrt(7)*sqrt(5)*(2*
x + 1) - 10296107698092025625520320851652379012251908564769041680092626442342953
147427423262647598*x - 514805384904601281276016042582618950612595428238452084004
6313221171476573713711631323799) - 369607*sqrt(5)*(87017964052957797337208059672
8111645575197008532950901616287851935257530573818652464500*sqrt(7)*sqrt(5) - 514
80538490460128127601604258261895061259542823845208400463132211714765737137116313
23799))/(87017964052957797337208059672811164557519700853295090161628785193525753
0573818652464500*sqrt(7)*sqrt(5) - 514805384904601281276016042582618950612595428
2384520840046313221171476573713711631323799)))/((250141922*sqrt(7)*(25*x^4 + 30*
x^3 + 29*x^2 + 12*x + 4) - 452768575*sqrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x +
4))*sqrt((250141922*sqrt(7)*sqrt(5) - 2263842875)/(32358971877628900*sqrt(7)*sqr
t(5) - 208999111504375959)))
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 x + 1} \left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1+2*x)**(1/2)/(5*x**2+3*x+2)**3,x)
[Out]
Integral(1/(sqrt(2*x + 1)*(5*x**2 + 3*x + 2)**3), x)
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3} \sqrt{2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x^2 + 3*x + 2)^3*sqrt(2*x + 1)),x, algorithm="giac")
[Out]
integrate(1/((5*x^2 + 3*x + 2)^3*sqrt(2*x + 1)), x)