3.2316 \(\int \frac{\sqrt{1+2 x}}{\left (2+3 x+5 x^2\right )^3} \, dx\)
Optimal. Leaf size=300 \[ \frac{\sqrt{2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}+\frac{\sqrt{2 x+1} (1790 x+599)}{13454 \left (5 x^2+3 x+2\right )}+\frac{\sqrt{\frac{1}{434} \left (1806875 \sqrt{35}-9651062\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{13454}-\frac{\sqrt{\frac{1}{434} \left (1806875 \sqrt{35}-9651062\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{13454}-\frac{\sqrt{\frac{1}{434} \left (9651062+1806875 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{6727}+\frac{\sqrt{\frac{1}{434} \left (9651062+1806875 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{6727} \]
[Out]
(Sqrt[1 + 2*x]*(3 + 10*x))/(62*(2 + 3*x + 5*x^2)^2) + (Sqrt[1 + 2*x]*(599 + 1790
*x))/(13454*(2 + 3*x + 5*x^2)) - (Sqrt[(9651062 + 1806875*Sqrt[35])/434]*ArcTan[
(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/6727 + (
Sqrt[(9651062 + 1806875*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt
[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/6727 + (Sqrt[(-9651062 + 1806875*Sqrt[35])
/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/13454
- (Sqrt[(-9651062 + 1806875*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35]
)]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/13454
_______________________________________________________________________________________
Rubi [A] time = 1.23297, antiderivative size = 300, 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
\[ \frac{\sqrt{2 x+1} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}+\frac{\sqrt{2 x+1} (1790 x+599)}{13454 \left (5 x^2+3 x+2\right )}+\frac{\sqrt{\frac{1}{434} \left (1806875 \sqrt{35}-9651062\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{13454}-\frac{\sqrt{\frac{1}{434} \left (1806875 \sqrt{35}-9651062\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{13454}-\frac{\sqrt{\frac{1}{434} \left (9651062+1806875 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{6727}+\frac{\sqrt{\frac{1}{434} \left (9651062+1806875 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{6727} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 + 2*x]/(2 + 3*x + 5*x^2)^3,x]
[Out]
(Sqrt[1 + 2*x]*(3 + 10*x))/(62*(2 + 3*x + 5*x^2)^2) + (Sqrt[1 + 2*x]*(599 + 1790
*x))/(13454*(2 + 3*x + 5*x^2)) - (Sqrt[(9651062 + 1806875*Sqrt[35])/434]*ArcTan[
(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/6727 + (
Sqrt[(9651062 + 1806875*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt
[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/6727 + (Sqrt[(-9651062 + 1806875*Sqrt[35])
/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/13454
- (Sqrt[(-9651062 + 1806875*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35]
)]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/13454
_______________________________________________________________________________________
Rubi in Sympy [A] time = 79.0283, size = 400, normalized size = 1.33 \[ \frac{\sqrt{2 x + 1} \left (10 x + 3\right )}{62 \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{\sqrt{2 x + 1} \left (1790 x + 599\right )}{13454 \left (5 x^{2} + 3 x + 2\right )} - \frac{\sqrt{14} \left (- 179 \sqrt{35} + 544\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{188356 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (- 179 \sqrt{35} + 544\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{188356 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 358 \sqrt{35} + 1088\right )}{10} + \frac{1088 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{94178 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- 358 \sqrt{35} + 1088\right )}{10} + \frac{1088 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{94178 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+2*x)**(1/2)/(5*x**2+3*x+2)**3,x)
[Out]
sqrt(2*x + 1)*(10*x + 3)/(62*(5*x**2 + 3*x + 2)**2) + sqrt(2*x + 1)*(1790*x + 59
9)/(13454*(5*x**2 + 3*x + 2)) - sqrt(14)*(-179*sqrt(35) + 544)*log(2*x - sqrt(10
)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(188356*sqrt(2 + sqrt(35)
)) + sqrt(14)*(-179*sqrt(35) + 544)*log(2*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2
*x + 1)/5 + 1 + sqrt(35)/5)/(188356*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)*sq
rt(2 + sqrt(35))*(-358*sqrt(35) + 1088)/10 + 1088*sqrt(10)*sqrt(2 + sqrt(35))/5)
*atan(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/
(94178*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)*sqrt(2 + sq
rt(35))*(-358*sqrt(35) + 1088)/10 + 1088*sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(sqr
t(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(94178*sq
rt(-2 + sqrt(35))*sqrt(2 + sqrt(35)))
_______________________________________________________________________________________
Mathematica [C] time = 1.07953, size = 159, normalized size = 0.53 \[ \frac{\frac{31 \sqrt{2 x+1} \left (8950 x^3+8365 x^2+7547 x+1849\right )}{2 \left (5 x^2+3 x+2\right )^2}+\frac{\left (5549-902 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{\sqrt{-\frac{1}{5} i \left (\sqrt{31}-2 i\right )}}+\frac{\left (5549+902 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{\sqrt{\frac{1}{5} i \left (\sqrt{31}+2 i\right )}}}{208537} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 + 2*x]/(2 + 3*x + 5*x^2)^3,x]
[Out]
((31*Sqrt[1 + 2*x]*(1849 + 7547*x + 8365*x^2 + 8950*x^3))/(2*(2 + 3*x + 5*x^2)^2
) + ((5549 - (902*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]])/Sqr
t[(-I/5)*(-2*I + Sqrt[31])] + ((5549 + (902*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/S
qrt[-2 + I*Sqrt[31]]])/Sqrt[(I/5)*(2*I + Sqrt[31])])/208537
_______________________________________________________________________________________
Maple [B] time = 0.7, size = 1406, normalized size = 4.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+2*x)^(1/2)/(5*x^2+3*x+2)^3,x)
[Out]
-5/2919518*(-2/21475*5^(1/2)*(-13012793430*5^(1/2)+6673227400*7^(1/2))/(-390+40*
5^(1/2)*7^(1/2))*(1+2*x)^(3/2)+1/107375/(-390+40*5^(1/2)*7^(1/2))*(-214587133600
*5^(1/2)+114637845000*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(1+2*x)-2/107375*(-14
1628999400*5^(1/2)*7^(1/2)+440433008400)/(-390+40*5^(1/2)*7^(1/2))*(1+2*x)^(1/2)
+1/107375*(-76332028500*7^(1/2)+54802482000*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))^2-807375/417074/(-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)+10326175/5839036/(-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))*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+424
32/47089/(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))*5^(1/2)*7^(1/2)-21760/6727/(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))+628785825/2919518/(27050*5^(1/2)*7^(1/2)-150820)^(1/2)*arcta
n(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)-260699875/208537/(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)+5/29195
18*(2/21475*5^(1/2)*(-13012793430*5^(1/2)+6673227400*7^(1/2))/(-390+40*5^(1/2)*7
^(1/2))*(1+2*x)^(3/2)+1/107375/(-390+40*5^(1/2)*7^(1/2))*(-214587133600*5^(1/2)+
114637845000*7^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(1+2*x)+2/107375*(-14162899940
0*5^(1/2)*7^(1/2)+440433008400)/(-390+40*5^(1/2)*7^(1/2))*(1+2*x)^(1/2)+1/107375
*(-76332028500*7^(1/2)+54802482000*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+807375/417074/(-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)-10326175/5839036/(-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)))*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+42432/4708
9/(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))/(270
50*5^(1/2)*7^(1/2)-150820)^(1/2))*5^(1/2)*7^(1/2)-21760/6727/(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)-15
0820)^(1/2))+628785825/2919518/(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)*5^(1/2)/(-100*5^(1/2)*7^(1/2)+975)*7^(1/2)-260699875/208537/(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. \[ \int \frac{\sqrt{2 \, x + 1}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x + 1)/(5*x^2 + 3*x + 2)^3,x, algorithm="maxima")
[Out]
integrate(sqrt(2*x + 1)/(5*x^2 + 3*x + 2)^3, x)
_______________________________________________________________________________________
Fricas [A] time = 0.280252, size = 1432, normalized size = 4.77 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x + 1)/(5*x^2 + 3*x + 2)^3,x, algorithm="fricas")
[Out]
1/331067502164*6727^(3/4)*sqrt(118)*sqrt(31)*(6727^(1/4)*sqrt(118)*sqrt(31)*(965
1062*sqrt(7)*(8950*x^3 + 8365*x^2 + 7547*x + 1849) - 12648125*sqrt(5)*(8950*x^3
+ 8365*x^2 + 7547*x + 1849))*sqrt(2*x + 1)*sqrt((9651062*sqrt(7)*sqrt(5) - 63240
625)/(34876525302500*sqrt(7)*sqrt(5) - 207410902024719)) + 102361876*17405^(1/4)
*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(89621*17405^(1/4)*sqrt(31)
*(4510*sqrt(7) - 7353*sqrt(5))/(sqrt(64015)*6727^(1/4)*sqrt(118)*sqrt(31)*(96510
62*sqrt(7) - 12648125*sqrt(5))*sqrt(sqrt(7)*(17405^(1/4)*6727^(1/4)*sqrt(118)*(1
642547770473647209400805601888436673318927832357191486387194818574245608221*sqrt
(7)*sqrt(5) - 971744365767374093275676241985898405111545411134073780427061985559
3870274356)*sqrt(2*x + 1)*sqrt((9651062*sqrt(7)*sqrt(5) - 63240625)/(34876525302
500*sqrt(7)*sqrt(5) - 207410902024719)) + 295*sqrt(7)*(1024684981551686367941599
233587743407156361843112686869844550959900512500*sqrt(7)*sqrt(5)*(2*x + 1) - 121
24236206810389108833247137616807372356055750881450605494515043445439198*x - 6062
118103405194554416623568808403686178027875440725302747257521722719599) + 413*sqr
t(5)*(1024684981551686367941599233587743407156361843112686869844550959900512500*
sqrt(7)*sqrt(5) - 60621181034051945544166235688084036861780278754407253027472575
21722719599))/(10246849815516863679415992335877434071563618431126868698445509599
00512500*sqrt(7)*sqrt(5) - 60621181034051945544166235688084036861780278754407253
02747257521722719599))*sqrt((9651062*sqrt(7)*sqrt(5) - 63240625)/(34876525302500
*sqrt(7)*sqrt(5) - 207410902024719)) + 64015*6727^(1/4)*sqrt(118)*sqrt(2*x + 1)*
(9651062*sqrt(7) - 12648125*sqrt(5))*sqrt((9651062*sqrt(7)*sqrt(5) - 63240625)/(
34876525302500*sqrt(7)*sqrt(5) - 207410902024719)) + 2778251*17405^(1/4)*(895*sq
rt(7) - 544*sqrt(5)))) + 102361876*17405^(1/4)*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2
+ 12*x + 4)*arctan(89621*17405^(1/4)*sqrt(31)*(4510*sqrt(7) - 7353*sqrt(5))/(sq
rt(64015)*6727^(1/4)*sqrt(118)*sqrt(31)*(9651062*sqrt(7) - 12648125*sqrt(5))*sqr
t(-sqrt(7)*(17405^(1/4)*6727^(1/4)*sqrt(118)*(1642547770473647209400805601888436
673318927832357191486387194818574245608221*sqrt(7)*sqrt(5) - 9717443657673740932
756762419858984051115454111340737804270619855593870274356)*sqrt(2*x + 1)*sqrt((9
651062*sqrt(7)*sqrt(5) - 63240625)/(34876525302500*sqrt(7)*sqrt(5) - 20741090202
4719)) - 295*sqrt(7)*(1024684981551686367941599233587743407156361843112686869844
550959900512500*sqrt(7)*sqrt(5)*(2*x + 1) - 121242362068103891088332471376168073
72356055750881450605494515043445439198*x - 6062118103405194554416623568808403686
178027875440725302747257521722719599) - 413*sqrt(5)*(102468498155168636794159923
3587743407156361843112686869844550959900512500*sqrt(7)*sqrt(5) - 606211810340519
4554416623568808403686178027875440725302747257521722719599))/(102468498155168636
7941599233587743407156361843112686869844550959900512500*sqrt(7)*sqrt(5) - 606211
8103405194554416623568808403686178027875440725302747257521722719599))*sqrt((9651
062*sqrt(7)*sqrt(5) - 63240625)/(34876525302500*sqrt(7)*sqrt(5) - 20741090202471
9)) + 64015*6727^(1/4)*sqrt(118)*sqrt(2*x + 1)*(9651062*sqrt(7) - 12648125*sqrt(
5))*sqrt((9651062*sqrt(7)*sqrt(5) - 63240625)/(34876525302500*sqrt(7)*sqrt(5) -
207410902024719)) - 2778251*17405^(1/4)*(895*sqrt(7) - 544*sqrt(5)))) + 17405^(1
/4)*sqrt(31)*(9651062*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 12648125*s
qrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*log(25005859375*sqrt(7)*(17405^(1/
4)*6727^(1/4)*sqrt(118)*(1642547770473647209400805601888436673318927832357191486
387194818574245608221*sqrt(7)*sqrt(5) - 9717443657673740932756762419858984051115
454111340737804270619855593870274356)*sqrt(2*x + 1)*sqrt((9651062*sqrt(7)*sqrt(5
) - 63240625)/(34876525302500*sqrt(7)*sqrt(5) - 207410902024719)) + 295*sqrt(7)*
(1024684981551686367941599233587743407156361843112686869844550959900512500*sqrt(
7)*sqrt(5)*(2*x + 1) - 121242362068103891088332471376168073723560557508814506054
94515043445439198*x - 6062118103405194554416623568808403686178027875440725302747
257521722719599) + 413*sqrt(5)*(102468498155168636794159923358774340715636184311
2686869844550959900512500*sqrt(7)*sqrt(5) - 606211810340519455441662356880840368
6178027875440725302747257521722719599))/(102468498155168636794159923358774340715
6361843112686869844550959900512500*sqrt(7)*sqrt(5) - 606211810340519455441662356
8808403686178027875440725302747257521722719599)) - 17405^(1/4)*sqrt(31)*(9651062
*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 12648125*sqrt(5)*(25*x^4 + 30*x
^3 + 29*x^2 + 12*x + 4))*log(-25005859375*sqrt(7)*(17405^(1/4)*6727^(1/4)*sqrt(1
18)*(164254777047364720940080560188843667331892783235719148638719481857424560822
1*sqrt(7)*sqrt(5) - 971744365767374093275676241985898405111545411134073780427061
9855593870274356)*sqrt(2*x + 1)*sqrt((9651062*sqrt(7)*sqrt(5) - 63240625)/(34876
525302500*sqrt(7)*sqrt(5) - 207410902024719)) - 295*sqrt(7)*(1024684981551686367
941599233587743407156361843112686869844550959900512500*sqrt(7)*sqrt(5)*(2*x + 1)
- 12124236206810389108833247137616807372356055750881450605494515043445439198*x
- 6062118103405194554416623568808403686178027875440725302747257521722719599) - 4
13*sqrt(5)*(10246849815516863679415992335877434071563618431126868698445509599005
12500*sqrt(7)*sqrt(5) - 60621181034051945544166235688084036861780278754407253027
47257521722719599))/(10246849815516863679415992335877434071563618431126868698445
50959900512500*sqrt(7)*sqrt(5) - 60621181034051945544166235688084036861780278754
40725302747257521722719599)))/((9651062*sqrt(7)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x
+ 4) - 12648125*sqrt(5)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*sqrt((9651062*sq
rt(7)*sqrt(5) - 63240625)/(34876525302500*sqrt(7)*sqrt(5) - 207410902024719)))
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Sympy [A] time = 22.7079, size = 199, normalized size = 0.66 \[ \frac{286400 \left (2 x + 1\right )^{\frac{7}{2}}}{- 24109568 x + 5381600 \left (2 x + 1\right )^{4} - 8610560 \left (2 x + 1\right )^{3} + 18512704 \left (2 x + 1\right )^{2} - 1506848} - \frac{323840 \left (2 x + 1\right )^{\frac{5}{2}}}{- 24109568 x + 5381600 \left (2 x + 1\right )^{4} - 8610560 \left (2 x + 1\right )^{3} + 18512704 \left (2 x + 1\right )^{2} - 1506848} + \frac{754496 \left (2 x + 1\right )^{\frac{3}{2}}}{- 24109568 x + 5381600 \left (2 x + 1\right )^{4} - 8610560 \left (2 x + 1\right )^{3} + 18512704 \left (2 x + 1\right )^{2} - 1506848} - \frac{243712 \sqrt{2 x + 1}}{- 24109568 x + 5381600 \left (2 x + 1\right )^{4} - 8610560 \left (2 x + 1\right )^{3} + 18512704 \left (2 x + 1\right )^{2} - 1506848} + 64 \operatorname{RootSum}{\left (75465931487403231630327808 t^{4} + 9053854476152406016 t^{2} + 333142578125, \left ( t \mapsto t \log{\left (\frac{21632117045402271744 t^{3}}{158378125} + \frac{10865340674816 t}{1108646875} + \sqrt{2 x + 1} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+2*x)**(1/2)/(5*x**2+3*x+2)**3,x)
[Out]
286400*(2*x + 1)**(7/2)/(-24109568*x + 5381600*(2*x + 1)**4 - 8610560*(2*x + 1)*
*3 + 18512704*(2*x + 1)**2 - 1506848) - 323840*(2*x + 1)**(5/2)/(-24109568*x + 5
381600*(2*x + 1)**4 - 8610560*(2*x + 1)**3 + 18512704*(2*x + 1)**2 - 1506848) +
754496*(2*x + 1)**(3/2)/(-24109568*x + 5381600*(2*x + 1)**4 - 8610560*(2*x + 1)*
*3 + 18512704*(2*x + 1)**2 - 1506848) - 243712*sqrt(2*x + 1)/(-24109568*x + 5381
600*(2*x + 1)**4 - 8610560*(2*x + 1)**3 + 18512704*(2*x + 1)**2 - 1506848) + 64*
RootSum(75465931487403231630327808*_t**4 + 9053854476152406016*_t**2 + 333142578
125, Lambda(_t, _t*log(21632117045402271744*_t**3/158378125 + 10865340674816*_t/
1108646875 + sqrt(2*x + 1))))
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{2 \, x + 1}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x + 1)/(5*x^2 + 3*x + 2)^3,x, algorithm="giac")
[Out]
integrate(sqrt(2*x + 1)/(5*x^2 + 3*x + 2)^3, x)