3.2302 \(\int \frac{1}{\sqrt{1+2 x} \left (2+3 x+5 x^2\right )} \, dx\)

Optimal. Leaf size=218 \[ -\frac{\log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{\sqrt{14 \left (2+\sqrt{35}\right )}}+\frac{\log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{\sqrt{14 \left (2+\sqrt{35}\right )}}-\sqrt{\frac{2}{217} \left (2+\sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\sqrt{\frac{2}{217} \left (2+\sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

[Out]

-(Sqrt[(2*(2 + Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x
])/Sqrt[10*(-2 + Sqrt[35])]]) + Sqrt[(2*(2 + Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2
+ Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]] - Log[Sqrt[35] - Sqrt
[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[14*(2 + Sqrt[35])] + Log[S
qrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[14*(2 + Sqrt
[35])]

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Rubi [A]  time = 0.704591, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{\sqrt{14 \left (2+\sqrt{35}\right )}}+\frac{\log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{\sqrt{14 \left (2+\sqrt{35}\right )}}-\sqrt{\frac{2}{217} \left (2+\sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\sqrt{\frac{2}{217} \left (2+\sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(2*(2 + Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x
])/Sqrt[10*(-2 + Sqrt[35])]]) + Sqrt[(2*(2 + Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2
+ Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]] - Log[Sqrt[35] - Sqrt
[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[14*(2 + Sqrt[35])] + Log[S
qrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[14*(2 + Sqrt
[35])]

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Rubi in Sympy [A]  time = 47.9049, size = 223, normalized size = 1.02 \[ - \frac{\sqrt{14} \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{14 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{14 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{7 \sqrt{-2 + \sqrt{35}}} + \frac{\sqrt{14} \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{7 \sqrt{-2 + \sqrt{35}}} \]

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),x)

[Out]

-sqrt(14)*log(2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5
)/(14*sqrt(2 + sqrt(35))) + sqrt(14)*log(2*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt(
2*x + 1)/5 + 1 + sqrt(35)/5)/(14*sqrt(2 + sqrt(35))) + sqrt(14)*atan(sqrt(10)*(s
qrt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(7*sqrt(-2 + sqrt
(35))) + sqrt(14)*atan(sqrt(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt
(-2 + sqrt(35)))/(7*sqrt(-2 + sqrt(35)))

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Mathematica [C]  time = 0.123467, size = 95, normalized size = 0.44 \[ \frac{2 i \left (\sqrt{-2-i \sqrt{31}} \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )-\sqrt{-2+i \sqrt{31}} \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )\right )}{\sqrt{217}} \]

Antiderivative was successfully verified.

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

[Out]

((2*I)*(-(Sqrt[-2 + I*Sqrt[31]]*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]]) +
Sqrt[-2 - I*Sqrt[31]]*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 + I*Sqrt[31]]]))/Sqrt[217]

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Maple [B]  time = 0.046, size = 607, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/62*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)+1/217*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)-
5/31/(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)+2/217/(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)+4
/7/(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)+1/62*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)-1/217*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)-5/31/(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)+2/217/(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)+4/7/(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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.250186, size = 832, normalized size = 3.82 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/417074*6727^(3/4)*sqrt(31)*sqrt(2)*(sqrt(31)*(2*sqrt(7)*5^(1/4) - 7*5^(3/4))*l
og(620/7*sqrt(7)*(6727^(1/4)*5^(1/4)*sqrt(2)*sqrt(2*x + 1)*sqrt((2*sqrt(7)*sqrt(
5) - 35)/(4*sqrt(7)*sqrt(5) - 39)) + 5*sqrt(7)*(2*x + 1) + 7*sqrt(5))) - sqrt(31
)*(2*sqrt(7)*5^(1/4) - 7*5^(3/4))*log(-620/7*sqrt(7)*(6727^(1/4)*5^(1/4)*sqrt(2)
*sqrt(2*x + 1)*sqrt((2*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sqrt(5) - 39)) - 5*sqrt(
7)*(2*x + 1) - 7*sqrt(5))) - 124*sqrt(7)*5^(1/4)*arctan(217*sqrt(31)*(5*sqrt(7)*
5^(1/4) - 2*5^(3/4))/(6727^(1/4)*sqrt(31)*sqrt(155/7)*sqrt(2)*sqrt(sqrt(7)*(6727
^(1/4)*5^(1/4)*sqrt(2)*sqrt(2*x + 1)*sqrt((2*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sq
rt(5) - 39)) + 5*sqrt(7)*(2*x + 1) + 7*sqrt(5)))*(2*sqrt(7) - 7*sqrt(5))*sqrt((2
*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sqrt(5) - 39)) + 155*6727^(1/4)*sqrt(2)*sqrt(2
*x + 1)*(2*sqrt(7) - 7*sqrt(5))*sqrt((2*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sqrt(5)
- 39)) - 6727*5^(3/4))) - 124*sqrt(7)*5^(1/4)*arctan(217*sqrt(31)*(5*sqrt(7)*5^
(1/4) - 2*5^(3/4))/(6727^(1/4)*sqrt(31)*sqrt(155/7)*sqrt(2)*sqrt(-sqrt(7)*(6727^
(1/4)*5^(1/4)*sqrt(2)*sqrt(2*x + 1)*sqrt((2*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sqr
t(5) - 39)) - 5*sqrt(7)*(2*x + 1) - 7*sqrt(5)))*(2*sqrt(7) - 7*sqrt(5))*sqrt((2*
sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sqrt(5) - 39)) + 155*6727^(1/4)*sqrt(2)*sqrt(2*
x + 1)*(2*sqrt(7) - 7*sqrt(5))*sqrt((2*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sqrt(5)
- 39)) + 6727*5^(3/4))))/((2*sqrt(7) - 7*sqrt(5))*sqrt((2*sqrt(7)*sqrt(5) - 35)/
(4*sqrt(7)*sqrt(5) - 39)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 x + 1} \left (5 x^{2} + 3 x + 2\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(2*x + 1)*(5*x**2 + 3*x + 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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