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

Optimal. Leaf size=11 \[ 2 \sqrt{x^2+x} \]

[Out]

2*Sqrt[x + x^2]

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Rubi [A]  time = 0.00654045, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ 2 \sqrt{x^2+x} \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x)/Sqrt[x + x^2],x]

[Out]

2*Sqrt[x + x^2]

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Rubi in Sympy [A]  time = 1.05087, size = 8, normalized size = 0.73 \[ 2 \sqrt{x^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x**2 + x)

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Mathematica [A]  time = 0.0138085, size = 11, normalized size = 1. \[ 2 \sqrt{x (x+1)} \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[x*(1 + x)]

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Maple [A]  time = 0.005, size = 14, normalized size = 1.3 \[ 2\,{\frac{x \left ( 1+x \right ) }{\sqrt{{x}^{2}+x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*x)/(x^2+x)^(1/2),x)

[Out]

2*x*(1+x)/(x^2+x)^(1/2)

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Maxima [A]  time = 0.705094, size = 12, normalized size = 1.09 \[ 2 \, \sqrt{x^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x^2 + x)

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Fricas [A]  time = 0.263172, size = 57, normalized size = 5.18 \[ -\frac{8 \, x^{2} - 2 \, \sqrt{x^{2} + x}{\left (4 \, x + 1\right )} + 6 \, x - 1}{2 \,{\left (2 \, x - 2 \, \sqrt{x^{2} + x} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(8*x^2 - 2*sqrt(x^2 + x)*(4*x + 1) + 6*x - 1)/(2*x - 2*sqrt(x^2 + x) + 1)

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Sympy [A]  time = 0.293673, size = 8, normalized size = 0.73 \[ 2 \sqrt{x^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x**2 + x)

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GIAC/XCAS [A]  time = 0.261638, size = 12, normalized size = 1.09 \[ 2 \, \sqrt{x^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x^2 + x)