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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 1.00 \[ 2 \sqrt {x (x+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x)/Sqrt[x + x^2],x]

[Out]

2*Sqrt[x*(1 + x)]

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fricas [A]  time = 0.53, size = 9, normalized size = 0.82 \[ 2 \, \sqrt {x^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x^2 + x)

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giac [A]  time = 0.36, size = 9, normalized size = 0.82 \[ 2 \, \sqrt {x^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x^2 + x)

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maple [A]  time = 0.00, size = 14, normalized size = 1.27 \[ \frac {2 \left (x +1\right ) x}{\sqrt {x^{2}+x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.63, size = 9, normalized size = 0.82 \[ 2 \, \sqrt {x^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x^2 + x)

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mupad [B]  time = 3.52, size = 9, normalized size = 0.82 \[ 2\,\sqrt {x\,\left (x+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 1)/(x + x^2)^(1/2),x)

[Out]

2*(x*(x + 1))^(1/2)

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sympy [A]  time = 0.14, 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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