3.2301 \(\int \frac{\sqrt{1+2 x}}{2+3 x+5 x^2} \, dx\)

Optimal. Leaf size=222 \[ \frac{\log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}-\frac{\log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}-\sqrt{\frac{2}{5 \left (\sqrt{35}-2\right )}} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\sqrt{\frac{2}{5 \left (\sqrt{35}-2\right )}} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

[Out]

-(Sqrt[2/(5*(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x]
)/Sqrt[10*(-2 + Sqrt[35])]]) + Sqrt[2/(5*(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 +
Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]] + Log[Sqrt[35] - Sqrt[1
0*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[10*(2 + Sqrt[35])] - Log[Sqr
t[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[10*(2 + Sqrt[3
5])]

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Rubi [A]  time = 0.690727, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}-\frac{\log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{\sqrt{10 \left (2+\sqrt{35}\right )}}-\sqrt{\frac{2}{5 \left (\sqrt{35}-2\right )}} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\sqrt{\frac{2}{5 \left (\sqrt{35}-2\right )}} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[2/(5*(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x]
)/Sqrt[10*(-2 + Sqrt[35])]]) + Sqrt[2/(5*(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 +
Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]] + Log[Sqrt[35] - Sqrt[1
0*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[10*(2 + Sqrt[35])] - Log[Sqr
t[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)]/Sqrt[10*(2 + Sqrt[3
5])]

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Rubi in Sympy [A]  time = 51.9155, size = 223, normalized size = 1. \[ \frac{\sqrt{10} \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{10 \sqrt{2 + \sqrt{35}}} - \frac{\sqrt{10} \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{10 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{10} \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{5 \sqrt{-2 + \sqrt{35}}} + \frac{\sqrt{10} \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{5 \sqrt{-2 + \sqrt{35}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(10)*log(2*x - sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)
/(10*sqrt(2 + sqrt(35))) - sqrt(10)*log(2*x + sqrt(10)*sqrt(2 + sqrt(35))*sqrt(2
*x + 1)/5 + 1 + sqrt(35)/5)/(10*sqrt(2 + sqrt(35))) + sqrt(10)*atan(sqrt(10)*(sq
rt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/(5*sqrt(-2 + sqrt(
35))) + sqrt(10)*atan(sqrt(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(
-2 + sqrt(35)))/(5*sqrt(-2 + sqrt(35)))

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Mathematica [C]  time = 0.288042, size = 112, normalized size = 0.5 \[ \frac{2 \left (\sqrt{-2+i \sqrt{31}} \left (\sqrt{31}-2 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )+\sqrt{-2-i \sqrt{31}} \left (\sqrt{31}+2 i\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )\right )}{5 \sqrt{217}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.064, size = 486, normalized size = 2.2 \[ -{\frac{\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5}}{155}\ln \left ( -\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5}\sqrt{1+2\,x}+\sqrt{5}\sqrt{7}+10\,x+5 \right ) }-{\frac{4\,\sqrt{5}\sqrt{7}+8}{31\,\sqrt{10\,\sqrt{5}\sqrt{7}-20}}\arctan \left ({\frac{1}{\sqrt{10\,\sqrt{5}\sqrt{7}-20}} \left ( -\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5}+10\,\sqrt{1+2\,x} \right ) } \right ) }+{\frac{\sqrt{7}\sqrt{2\,\sqrt{5}\sqrt{7}+4}}{62}\ln \left ( -\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5}\sqrt{1+2\,x}+\sqrt{5}\sqrt{7}+10\,x+5 \right ) }+{\frac{ \left ( 2\,\sqrt{5}\sqrt{7}+4 \right ) \sqrt{5}\sqrt{7}}{31\,\sqrt{10\,\sqrt{5}\sqrt{7}-20}}\arctan \left ({\frac{1}{\sqrt{10\,\sqrt{5}\sqrt{7}-20}} \left ( -\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5}+10\,\sqrt{1+2\,x} \right ) } \right ) }+{\frac{\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5}}{155}\ln \left ( \sqrt{5}\sqrt{7}+10\,x+5+\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5}\sqrt{1+2\,x} \right ) }-{\frac{4\,\sqrt{5}\sqrt{7}+8}{31\,\sqrt{10\,\sqrt{5}\sqrt{7}-20}}\arctan \left ({\frac{1}{\sqrt{10\,\sqrt{5}\sqrt{7}-20}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5} \right ) } \right ) }-{\frac{\sqrt{7}\sqrt{2\,\sqrt{5}\sqrt{7}+4}}{62}\ln \left ( \sqrt{5}\sqrt{7}+10\,x+5+\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5}\sqrt{1+2\,x} \right ) }+{\frac{ \left ( 2\,\sqrt{5}\sqrt{7}+4 \right ) \sqrt{5}\sqrt{7}}{31\,\sqrt{10\,\sqrt{5}\sqrt{7}-20}}\arctan \left ({\frac{1}{\sqrt{10\,\sqrt{5}\sqrt{7}-20}} \left ( 10\,\sqrt{1+2\,x}+\sqrt{2\,\sqrt{5}\sqrt{7}+4}\sqrt{5} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/155*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)-2/31/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arct
an((-(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)-2
0)^(1/2))*(2*5^(1/2)*7^(1/2)+4)+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)*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+1/31/(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)*5^(1/2)*7^(1/2)+1/
155*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)-2/31/(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)-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))*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+1/31/(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)*5^(1/2)*7^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.25593, size = 1083, normalized size = 4.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/297910*4805^(3/4)*sqrt(31)*sqrt(2)*(sqrt(31)*7^(1/4)*(5*sqrt(7) - 2*sqrt(5))*l
og(124/175*sqrt(5)*(4805^(1/4)*7^(1/4)*sqrt(2)*(2081*sqrt(7)*sqrt(5) - 10920)*sq
rt(2*x + 1)*sqrt((2*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sqrt(5) - 39)) + 35*sqrt(5)
*(312*sqrt(7)*sqrt(5)*(2*x + 1) - 4162*x - 2081) + 35*sqrt(7)*(312*sqrt(7)*sqrt(
5) - 2081))/(312*sqrt(7)*sqrt(5) - 2081)) - sqrt(31)*7^(1/4)*(5*sqrt(7) - 2*sqrt
(5))*log(-124/175*sqrt(5)*(4805^(1/4)*7^(1/4)*sqrt(2)*(2081*sqrt(7)*sqrt(5) - 10
920)*sqrt(2*x + 1)*sqrt((2*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sqrt(5) - 39)) - 35*
sqrt(5)*(312*sqrt(7)*sqrt(5)*(2*x + 1) - 4162*x - 2081) - 35*sqrt(7)*(312*sqrt(7
)*sqrt(5) - 2081))/(312*sqrt(7)*sqrt(5) - 2081)) - 124*7^(1/4)*sqrt(5)*arctan(15
5*sqrt(31)*7^(1/4)*(5*sqrt(7) - 2*sqrt(5))/(4805^(1/4)*sqrt(31)*sqrt(31/7)*sqrt(
2)*(5*sqrt(7) - 2*sqrt(5))*sqrt(sqrt(5)*(4805^(1/4)*7^(1/4)*sqrt(2)*(2081*sqrt(7
)*sqrt(5) - 10920)*sqrt(2*x + 1)*sqrt((2*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sqrt(5
) - 39)) + 35*sqrt(5)*(312*sqrt(7)*sqrt(5)*(2*x + 1) - 4162*x - 2081) + 35*sqrt(
7)*(312*sqrt(7)*sqrt(5) - 2081))/(312*sqrt(7)*sqrt(5) - 2081))*sqrt((2*sqrt(7)*s
qrt(5) - 35)/(4*sqrt(7)*sqrt(5) - 39)) + 155*4805^(1/4)*sqrt(2)*sqrt(2*x + 1)*(5
*sqrt(7) - 2*sqrt(5))*sqrt((2*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sqrt(5) - 39)) -
4805*7^(1/4)*sqrt(5))) - 124*7^(1/4)*sqrt(5)*arctan(155*sqrt(31)*7^(1/4)*(5*sqrt
(7) - 2*sqrt(5))/(4805^(1/4)*sqrt(31)*sqrt(31/7)*sqrt(2)*(5*sqrt(7) - 2*sqrt(5))
*sqrt(-sqrt(5)*(4805^(1/4)*7^(1/4)*sqrt(2)*(2081*sqrt(7)*sqrt(5) - 10920)*sqrt(2
*x + 1)*sqrt((2*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sqrt(5) - 39)) - 35*sqrt(5)*(31
2*sqrt(7)*sqrt(5)*(2*x + 1) - 4162*x - 2081) - 35*sqrt(7)*(312*sqrt(7)*sqrt(5) -
2081))/(312*sqrt(7)*sqrt(5) - 2081))*sqrt((2*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*s
qrt(5) - 39)) + 155*4805^(1/4)*sqrt(2)*sqrt(2*x + 1)*(5*sqrt(7) - 2*sqrt(5))*sqr
t((2*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sqrt(5) - 39)) + 4805*7^(1/4)*sqrt(5))))/(
(5*sqrt(7) - 2*sqrt(5))*sqrt((2*sqrt(7)*sqrt(5) - 35)/(4*sqrt(7)*sqrt(5) - 39)))

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Sympy [A]  time = 5.46564, size = 32, normalized size = 0.14 \[ 4 \operatorname{RootSum}{\left (1230080 t^{4} + 1984 t^{2} + 7, \left ( t \mapsto t \log{\left (9920 t^{3} + 8 t + \sqrt{2 x + 1} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*RootSum(1230080*_t**4 + 1984*_t**2 + 7, Lambda(_t, _t*log(9920*_t**3 + 8*_t +
sqrt(2*x + 1))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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