3.785 \(\int \frac{1}{(1+2 x) \sqrt{x+x^2}} \, dx\)

Optimal. Leaf size=12 \[ \tan ^{-1}\left (2 \sqrt{x^2+x}\right ) \]

[Out]

ArcTan[2*Sqrt[x + x^2]]

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Rubi [A]  time = 0.0241491, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \tan ^{-1}\left (2 \sqrt{x^2+x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[2*Sqrt[x + x^2]]

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Rubi in Sympy [A]  time = 2.24703, size = 10, normalized size = 0.83 \[ \operatorname{atan}{\left (2 \sqrt{x^{2} + x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(2*sqrt(x**2 + x))

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Mathematica [B]  time = 0.026756, size = 37, normalized size = 3.08 \[ \frac{2 \sqrt{x} \sqrt{x+1} \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+1}}\right )}{\sqrt{x (x+1)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 15, normalized size = 1.3 \[ -\arctan \left ({\frac{1}{\sqrt{4\, \left ( x+1/2 \right ) ^{2}-1}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+2*x)/(x^2+x)^(1/2),x)

[Out]

-arctan(1/(4*(x+1/2)^2-1)^(1/2))

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Maxima [A]  time = 0.773137, size = 15, normalized size = 1.25 \[ -\arcsin \left (\frac{1}{{\left | 2 \, x + 1 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arcsin(1/abs(2*x + 1))

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Fricas [A]  time = 0.265038, size = 23, normalized size = 1.92 \[ 2 \, \arctan \left (-2 \, x + 2 \, \sqrt{x^{2} + x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(-2*x + 2*sqrt(x^2 + x) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.264439, size = 23, normalized size = 1.92 \[ 2 \, \arctan \left (-2 \, x + 2 \, \sqrt{x^{2} + x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(-2*x + 2*sqrt(x^2 + x) - 1)