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

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

[Out]

2*Sqrt[x + x^2]

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Rubi [A]  time = 0.0030018, 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, Rules used = {629} \[ 2 \sqrt{x^2+x} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x)/Sqrt[x + x^2],x]

[Out]

2*Sqrt[x + x^2]

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{1+2 x}{\sqrt{x+x^2}} \, dx &=2 \sqrt{x+x^2}\\ \end{align*}

Mathematica [A]  time = 0.0054137, 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 = 0.003, size = 14, normalized size = 1.3 \begin{align*} 2\,{\frac{x \left ( 1+x \right ) }{\sqrt{{x}^{2}+x}}} \end{align*}

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 = 0.980865, size = 12, normalized size = 1.09 \begin{align*} 2 \, \sqrt{x^{2} + x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)/(x^2+x)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(x^2 + x)

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Fricas [A]  time = 1.42915, size = 23, normalized size = 2.09 \begin{align*} 2 \, \sqrt{x^{2} + x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)/(x^2+x)^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(x^2 + x)

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Sympy [A]  time = 0.123541, size = 8, normalized size = 0.73 \begin{align*} 2 \sqrt{x^{2} + x} \end{align*}

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 [A]  time = 1.09767, size = 12, normalized size = 1.09 \begin{align*} 2 \, \sqrt{x^{2} + x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)/(x^2+x)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(x^2 + x)