3.2314 \(\int \frac{(1+2 x)^{5/2}}{\left (2+3 x+5 x^2\right )^3} \, dx\)
Optimal. Leaf size=300 \[ -\frac{(5-4 x) (2 x+1)^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac{3 (78 x+11) \sqrt{2 x+1}}{1922 \left (5 x^2+3 x+2\right )}+\frac{3 \sqrt{\frac{1}{310} \left (2705 \sqrt{35}-15082\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1922}-\frac{3 \sqrt{\frac{1}{310} \left (2705 \sqrt{35}-15082\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1922}-\frac{3}{961} \sqrt{\frac{1}{310} \left (15082+2705 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{3}{961} \sqrt{\frac{1}{310} \left (15082+2705 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]
[Out]
-((5 - 4*x)*(1 + 2*x)^(3/2))/(62*(2 + 3*x + 5*x^2)^2) + (3*Sqrt[1 + 2*x]*(11 + 7
8*x))/(1922*(2 + 3*x + 5*x^2)) - (3*Sqrt[(15082 + 2705*Sqrt[35])/310]*ArcTan[(Sq
rt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/961 + (3*Sq
rt[(15082 + 2705*Sqrt[35])/310]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*
x])/Sqrt[10*(-2 + Sqrt[35])]])/961 + (3*Sqrt[(-15082 + 2705*Sqrt[35])/310]*Log[S
qrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/1922 - (3*Sqrt[(
-15082 + 2705*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x
] + 5*(1 + 2*x)])/1922
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Rubi [A] time = 1.29107, 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
\[ -\frac{(5-4 x) (2 x+1)^{3/2}}{62 \left (5 x^2+3 x+2\right )^2}+\frac{3 (78 x+11) \sqrt{2 x+1}}{1922 \left (5 x^2+3 x+2\right )}+\frac{3 \sqrt{\frac{1}{310} \left (2705 \sqrt{35}-15082\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1922}-\frac{3 \sqrt{\frac{1}{310} \left (2705 \sqrt{35}-15082\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{1922}-\frac{3}{961} \sqrt{\frac{1}{310} \left (15082+2705 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{3}{961} \sqrt{\frac{1}{310} \left (15082+2705 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + 2*x)^(5/2)/(2 + 3*x + 5*x^2)^3,x]
[Out]
-((5 - 4*x)*(1 + 2*x)^(3/2))/(62*(2 + 3*x + 5*x^2)^2) + (3*Sqrt[1 + 2*x]*(11 + 7
8*x))/(1922*(2 + 3*x + 5*x^2)) - (3*Sqrt[(15082 + 2705*Sqrt[35])/310]*ArcTan[(Sq
rt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/961 + (3*Sq
rt[(15082 + 2705*Sqrt[35])/310]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*
x])/Sqrt[10*(-2 + Sqrt[35])]])/961 + (3*Sqrt[(-15082 + 2705*Sqrt[35])/310]*Log[S
qrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/1922 - (3*Sqrt[(
-15082 + 2705*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x
] + 5*(1 + 2*x)])/1922
_______________________________________________________________________________________
Rubi in Sympy [A] time = 79.6311, size = 406, normalized size = 1.35 \[ - \frac{\left (- 4 x + 5\right ) \left (2 x + 1\right )^{\frac{3}{2}}}{62 \left (5 x^{2} + 3 x + 2\right )^{2}} + \frac{\sqrt{2 x + 1} \left (234 x + 33\right )}{1922 \left (5 x^{2} + 3 x + 2\right )} - \frac{\sqrt{14} \left (- \frac{117 \sqrt{35}}{5} + 84\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{26908 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (- \frac{117 \sqrt{35}}{5} + 84\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{26908 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- \frac{234 \sqrt{35}}{5} + 168\right )}{10} + \frac{168 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{13454 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- \frac{234 \sqrt{35}}{5} + 168\right )}{10} + \frac{168 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{13454 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+2*x)**(5/2)/(5*x**2+3*x+2)**3,x)
[Out]
-(-4*x + 5)*(2*x + 1)**(3/2)/(62*(5*x**2 + 3*x + 2)**2) + sqrt(2*x + 1)*(234*x +
33)/(1922*(5*x**2 + 3*x + 2)) - sqrt(14)*(-117*sqrt(35)/5 + 84)*log(2*x - sqrt(
10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(26908*sqrt(2 + sqrt(35
))) + sqrt(14)*(-117*sqrt(35)/5 + 84)*log(2*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt
(2*x + 1)/5 + 1 + sqrt(35)/5)/(26908*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)*s
qrt(2 + sqrt(35))*(-234*sqrt(35)/5 + 168)/10 + 168*sqrt(10)*sqrt(2 + sqrt(35))/5
)*atan(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))
/(13454*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)*sqrt(2 + s
qrt(35))*(-234*sqrt(35)/5 + 168)/10 + 168*sqrt(10)*sqrt(2 + sqrt(35))/5)*atan(sq
rt(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(13454*s
qrt(-2 + sqrt(35))*sqrt(2 + sqrt(35)))
_______________________________________________________________________________________
Mathematica [C] time = 1.11398, size = 155, normalized size = 0.52 \[ \frac{\sqrt{2 x+1} \left (1170 x^3+1115 x^2+381 x-89\right )}{1922 \left (5 x^2+3 x+2\right )^2}+\frac{3 \left (1209-218 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{29791 \sqrt{-10-5 i \sqrt{31}}}+\frac{3 \left (1209+218 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{29791 \sqrt{5 i \left (\sqrt{31}+2 i\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + 2*x)^(5/2)/(2 + 3*x + 5*x^2)^3,x]
[Out]
(Sqrt[1 + 2*x]*(-89 + 381*x + 1115*x^2 + 1170*x^3))/(1922*(2 + 3*x + 5*x^2)^2) +
(3*(1209 - (218*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]])/(297
91*Sqrt[-10 - (5*I)*Sqrt[31]]) + (3*(1209 + (218*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10
*x]/Sqrt[-2 + I*Sqrt[31]]])/(29791*Sqrt[(5*I)*(2*I + Sqrt[31])])
_______________________________________________________________________________________
Maple [B] time = 0.057, size = 662, normalized size = 2.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+2*x)^(5/2)/(5*x^2+3*x+2)^3,x)
[Out]
1600*(117/153760*(1+2*x)^(7/2)-4/4805*(1+2*x)^(5/2)+287/768800*(1+2*x)^(3/2)-147
/192200*(1+2*x)^(1/2))/(5*(1+2*x)^2-8*x+3)^2-327/297910*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)+141/119164*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)-327/29791/(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))/(1
0*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)+141/59582/(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)+24/961/(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)+327/297910*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)-141/119164*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)-327/29791/(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)+141/59582/(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)+24/961/(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. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 1)^(5/2)/(5*x^2 + 3*x + 2)^3,x, algorithm="maxima")
[Out]
integrate((2*x + 1)^(5/2)/(5*x^2 + 3*x + 2)^3, x)
_______________________________________________________________________________________
Fricas [A] time = 0.261802, size = 1184, normalized size = 3.95 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 1)^(5/2)/(5*x^2 + 3*x + 2)^3,x, algorithm="fricas")
[Out]
1/322338620*sqrt(5410)*(31*sqrt(5410)*(17645940*x^3 + 16816430*x^2 - 2705*sqrt(3
5)*(1170*x^3 + 1115*x^2 + 381*x - 89) + 5746242*x - 1342298)*sqrt(2*x + 1)*sqrt(
(15082*sqrt(35) - 94675)/(81593620*sqrt(35) - 483562599)) + 357492*10243835^(1/4
)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(2705*10243835^(1/4)*(47*sqrt(35)*
sqrt(31) - 218*sqrt(31))/(sqrt(83855)*sqrt(5410)*(2705*sqrt(35) - 15082)*sqrt((1
0243835^(1/4)*sqrt(5410)*(159078508844737*sqrt(35)*sqrt(31) - 941109169572002*sq
rt(31))*sqrt(2*x + 1)*sqrt((15082*sqrt(35) - 94675)/(81593620*sqrt(35) - 4835625
99)) + 40623600982826690*sqrt(35)*(2*x + 1) + 541*sqrt(35)*(15017967091618*sqrt(
35) - 88852113249725) - 480689932681012250*x - 240344966340506125)/(150179670916
18*sqrt(35) - 88852113249725))*sqrt((15082*sqrt(35) - 94675)/(81593620*sqrt(35)
- 483562599)) + 2705*sqrt(5410)*(2705*sqrt(35)*sqrt(31) - 15082*sqrt(31))*sqrt(2
*x + 1)*sqrt((15082*sqrt(35) - 94675)/(81593620*sqrt(35) - 483562599)) + 83855*1
0243835^(1/4)*(4*sqrt(35) - 39))) + 357492*10243835^(1/4)*(25*x^4 + 30*x^3 + 29*
x^2 + 12*x + 4)*arctan(2705*10243835^(1/4)*(47*sqrt(35)*sqrt(31) - 218*sqrt(31))
/(sqrt(83855)*sqrt(5410)*(2705*sqrt(35) - 15082)*sqrt(-(10243835^(1/4)*sqrt(5410
)*(159078508844737*sqrt(35)*sqrt(31) - 941109169572002*sqrt(31))*sqrt(2*x + 1)*s
qrt((15082*sqrt(35) - 94675)/(81593620*sqrt(35) - 483562599)) - 4062360098282669
0*sqrt(35)*(2*x + 1) - 541*sqrt(35)*(15017967091618*sqrt(35) - 88852113249725) +
480689932681012250*x + 240344966340506125)/(15017967091618*sqrt(35) - 888521132
49725))*sqrt((15082*sqrt(35) - 94675)/(81593620*sqrt(35) - 483562599)) + 2705*sq
rt(5410)*(2705*sqrt(35)*sqrt(31) - 15082*sqrt(31))*sqrt(2*x + 1)*sqrt((15082*sqr
t(35) - 94675)/(81593620*sqrt(35) - 483562599)) - 83855*10243835^(1/4)*(4*sqrt(3
5) - 39))) - 3*10243835^(1/4)*(2705*sqrt(35)*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2
+ 12*x + 4) - 15082*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*log(754695*(
10243835^(1/4)*sqrt(5410)*(159078508844737*sqrt(35)*sqrt(31) - 941109169572002*s
qrt(31))*sqrt(2*x + 1)*sqrt((15082*sqrt(35) - 94675)/(81593620*sqrt(35) - 483562
599)) + 40623600982826690*sqrt(35)*(2*x + 1) + 541*sqrt(35)*(15017967091618*sqrt
(35) - 88852113249725) - 480689932681012250*x - 240344966340506125)/(15017967091
618*sqrt(35) - 88852113249725)) + 3*10243835^(1/4)*(2705*sqrt(35)*sqrt(31)*(25*x
^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 15082*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12
*x + 4))*log(-754695*(10243835^(1/4)*sqrt(5410)*(159078508844737*sqrt(35)*sqrt(3
1) - 941109169572002*sqrt(31))*sqrt(2*x + 1)*sqrt((15082*sqrt(35) - 94675)/(8159
3620*sqrt(35) - 483562599)) - 40623600982826690*sqrt(35)*(2*x + 1) - 541*sqrt(35
)*(15017967091618*sqrt(35) - 88852113249725) + 480689932681012250*x + 2403449663
40506125)/(15017967091618*sqrt(35) - 88852113249725)))/((377050*x^4 + 452460*x^3
+ 437378*x^2 - 2705*sqrt(35)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) + 180984*x +
60328)*sqrt((15082*sqrt(35) - 94675)/(81593620*sqrt(35) - 483562599)))
_______________________________________________________________________________________
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+2*x)**(5/2)/(5*x**2+3*x+2)**3,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 1)^(5/2)/(5*x^2 + 3*x + 2)^3,x, algorithm="giac")
[Out]
integrate((2*x + 1)^(5/2)/(5*x^2 + 3*x + 2)^3, x)