3.2574 \(\int \frac{5+\sqrt{35}+10 x}{\sqrt{1+2 x} \left (2+3 x+5 x^2\right )} \, dx\)

Optimal. Leaf size=105 \[ 2 \sqrt{\frac{10}{\sqrt{35}-2}} \tan ^{-1}\left (\frac{\sqrt{20 x+10}+\sqrt{2+\sqrt{35}}}{\sqrt{\sqrt{35}-2}}\right )-2 \sqrt{\frac{10}{\sqrt{35}-2}} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{35}}-\sqrt{20 x+10}}{\sqrt{\sqrt{35}-2}}\right ) \]

[Out]

-2*Sqrt[10/(-2 + Sqrt[35])]*ArcTan[(Sqrt[2 + Sqrt[35]] - Sqrt[10 + 20*x])/Sqrt[-
2 + Sqrt[35]]] + 2*Sqrt[10/(-2 + Sqrt[35])]*ArcTan[(Sqrt[2 + Sqrt[35]] + Sqrt[10
+ 20*x])/Sqrt[-2 + Sqrt[35]]]

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Rubi [A]  time = 0.517886, antiderivative size = 115, normalized size of antiderivative = 1.1, number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ 2 \sqrt{\frac{10}{\sqrt{35}-2}} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-2 \sqrt{\frac{10}{\sqrt{35}-2}} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(5 + Sqrt[35] + 10*x)/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)),x]

[Out]

-2*Sqrt[10/(-2 + Sqrt[35])]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/
Sqrt[10*(-2 + Sqrt[35])]] + 2*Sqrt[10/(-2 + Sqrt[35])]*ArcTan[(Sqrt[10*(2 + Sqrt
[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]]

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Rubi in Sympy [A]  time = 24.7061, size = 110, normalized size = 1.05 \[ \frac{2 \sqrt{10} \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{\sqrt{-2 + \sqrt{35}}} + \frac{2 \sqrt{10} \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{\sqrt{-2 + \sqrt{35}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5+10*x+35**(1/2))/(5*x**2+3*x+2)/(1+2*x)**(1/2),x)

[Out]

2*sqrt(10)*atan(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + s
qrt(35)))/sqrt(-2 + sqrt(35)) + 2*sqrt(10)*atan(sqrt(10)*(sqrt(2*x + 1) + sqrt(2
0 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(35)))/sqrt(-2 + sqrt(35))

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Mathematica [C]  time = 0.348279, size = 130, normalized size = 1.24 \[ 2 \sqrt{\frac{5}{31}} \left (\frac{\left (-2 i+\sqrt{31}-i \sqrt{35}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{\sqrt{-2-i \sqrt{31}}}+\frac{\left (2 i+\sqrt{31}+i \sqrt{35}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{\sqrt{-2+i \sqrt{31}}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(5 + Sqrt[35] + 10*x)/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)),x]

[Out]

2*Sqrt[5/31]*(((-2*I + Sqrt[31] - I*Sqrt[35])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*
Sqrt[31]]])/Sqrt[-2 - I*Sqrt[31]] + ((2*I + Sqrt[31] + I*Sqrt[35])*ArcTan[Sqrt[5
+ 10*x]/Sqrt[-2 + I*Sqrt[31]]])/Sqrt[-2 + I*Sqrt[31]])

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Maple [A]  time = 0.888, size = 111, normalized size = 1.1 \[ 20\,{\frac{1}{\sqrt{10\,\sqrt{5}\sqrt{7}-20}}\arctan \left ({\frac{10\,\sqrt{1+2\,x}-\sqrt{5}\sqrt{2\,\sqrt{5}\sqrt{7}+4}}{\sqrt{10\,\sqrt{5}\sqrt{7}-20}}} \right ) }+20\,{\frac{1}{\sqrt{10\,\sqrt{5}\sqrt{7}-20}}\arctan \left ({\frac{10\,\sqrt{1+2\,x}+\sqrt{5}\sqrt{2\,\sqrt{5}\sqrt{7}+4}}{\sqrt{10\,\sqrt{5}\sqrt{7}-20}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5+10*x+35^(1/2))/(5*x^2+3*x+2)/(1+2*x)^(1/2),x)

[Out]

20/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)-5^(1/2)*(2*5^(1/2)*7^(
1/2)+4)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))+20/(10*5^(1/2)*7^(1/2)-20)^(1/2)*a
rctan((10*(1+2*x)^(1/2)+5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2))/(10*5^(1/2)*7^(1/2)
-20)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{10 \, x + \sqrt{35} + 5}{{\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt{2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((10*x + sqrt(35) + 5)/((5*x^2 + 3*x + 2)*sqrt(2*x + 1)),x, algorithm="maxima")

[Out]

integrate((10*x + sqrt(35) + 5)/((5*x^2 + 3*x + 2)*sqrt(2*x + 1)), x)

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Fricas [A]  time = 0.299866, size = 85, normalized size = 0.81 \[ \frac{2}{31} \, \sqrt{31} \sqrt{10 \, \sqrt{35} + 20} \arctan \left (\frac{\sqrt{35} \sqrt{31}{\left (40 \, x - 19\right )} + 5 \, \sqrt{31}{\left (78 \, x + 11\right )}}{31 \, \sqrt{2 \, x + 1} \sqrt{10 \, \sqrt{35} + 20}{\left (\sqrt{35} + 2\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((10*x + sqrt(35) + 5)/((5*x^2 + 3*x + 2)*sqrt(2*x + 1)),x, algorithm="fricas")

[Out]

2/31*sqrt(31)*sqrt(10*sqrt(35) + 20)*arctan(1/31*(sqrt(35)*sqrt(31)*(40*x - 19)
+ 5*sqrt(31)*(78*x + 11))/(sqrt(2*x + 1)*sqrt(10*sqrt(35) + 20)*(sqrt(35) + 2)))

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Sympy [A]  time = 12.7707, size = 0, normalized size = 0. \[ \mathrm{NaN} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5+10*x+35**(1/2))/(5*x**2+3*x+2)/(1+2*x)**(1/2),x)

[Out]

nan

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{10 \, x + \sqrt{35} + 5}{{\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt{2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((10*x + sqrt(35) + 5)/((5*x^2 + 3*x + 2)*sqrt(2*x + 1)),x, algorithm="giac")

[Out]

integrate((10*x + sqrt(35) + 5)/((5*x^2 + 3*x + 2)*sqrt(2*x + 1)), x)